Force 7. Force Control

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4 164 Part A Robotics Foundations loops closed around inner motion loops, typically available in industrial robots [7.2]. If an accurate model of the environment is not available, the force control action and the motion control action can be superimposed, resulting in a parallel force/position control scheme. In this approach, the force controller is designed so as to dominate the motion controller; hence, a position error would be tolerated along the constrained task directions in order to ensure force regulation [7.19]. 7.2 Indirect Force Control Part A 7.2 To gain insight into the problems arising at the interaction between the end-effector of a robot manipulator and the environment, it is worth analyzing the effects of a motion control strategy in the presence of a contact force and moment. To this aim, assume that a reference frame Σ e is attached to the end-effector, and let p e denote the position vector of the origin and R e the rotation matrix with respect to a fixed base frame. The end-effector velocity is denoted by the 6 1 twist vector v e = ( ṗ ) e ω e where ṗ e is the translational velocity and ω e the angular velocity, and can be computed from the n 1 joint velocity vector q using the linear mapping v e = J(q) q. (7.1) The matrix J is the 6 n end-effector geometric Jacobian. For simplicity, the case of nonredundant nonsingular manipulators is considered; therefore, n = 6 and the Jacobian is a square nonsingular matrix. The force f e and moment m e applied by the end-effector to the environment are the components of the wrench h e = ( f e m ). e It is useful to consider the operational space formulation of the dynamic model of a rigid robot manipulator in contact with the environment Λ(q) v e + Γ (q, q)v e + η(q) = h c h e, (7.2) where Λ(q) = (JH(q) 1 J ) 1 is the 6 6 operational space inertia matrix, Γ (q, q) = J C(q, q)j 1 Λ(q) JJ 1 is the wrench including centrifugal and Coriolis effects, and η(q) = J g(q) is the wrench of the gravitational effects; H(q), C(q, q) andg(q) are the corresponding quantities defined in the joint space. The vector h c = J τ is the equivalent end-effector wrench corresponding to the input joint torques τ Stiffness Control In the classical operational space formulation, the endeffector position and orientation is described by a 6 1 vector x e = ( p e ϕ ), e where ϕe is a set of Euler angles extracted from R e. Hence, a desired end-effector position and orientation can be assigned in terms of a vector x d, corresponding to the position of the origin p d and the rotation matrix R d of a desired frame Σ d. The endeffector error can be denoted as Δx de = x d x e,andthe corresponding velocity error, assuming a constant x d, can be expressed as Δẋ de = ẋ e = A 1 (ϕ e )v e, with ( ) I 0 A(ϕ e ) =, 0 T(ϕ e ) where I is the 3 3 identity matrix, 0 is a 3 3 null matrix, and T is the 3 3 matrix of the mapping ω e = T(ϕ e ) ϕ e, depending on the particular choice of the Euler angles. Consider the motion control law h c = A (ϕ e )K P Δx de K D v e + η(q), (7.3) corresponding to a simple PD + gravity compensation control in the operational space, where K P and K D are symmetric and positive-definite 6 6 matrices. In the absence of interaction with the environment (i. e., when h e = 0), the equilibrium v e = 0, Δx de = 0 for the closed-loop system, corresponding to the desired position and orientation for the end-effector, is asymptotically stable. The stability proof is based on the positive-definite Lyapunov function V = 1 2 v e Λ(q)v e Δx dek P Δx de, whose time derivative along the trajectories of the closed-loop system is the negative semidefinite function V = v e K Dv e. (7.4) In the presence of a constant wrench h e, using a similar Lyapunov argument, a different asymptotically stable equilibrium can be found, with a nonnull Δx de. The new equilibrium is the solution of the equation A (ϕ e )K P Δx de h e = 0, which can be written in the form Δx de = K 1 P A (ϕ e )h e, (7.5)

5 Force Control 7.2 Indirect Force Control 165 or, equivalently, as h e = A (ϕ e )K P Δx de. (7.6) Equation (7.6) shows that in the steady state the endeffector, under a proportional control action on the position and orientation error, behaves as a six-degreeof-freedom (DOF) spring in respect of the external force and moment h e. Thus, the matrix K P plays the role of an active stiffness, eaning that it is possible to act on the elements of K P so as to ensure a suitable elastic behavior of the end-effector during the interaction. Analogously, (7.5) represents a compliance relationship, where the matrix K 1 P plays the role of an active compliance. This approach, consisting of assigning a desired position and orientation and a suitable static relationship between the deviation of the end-effector position and orientation from the desired motion and the force exerted on the environment, is known as stiffness control. The selection of the stiffness/compliance parameters is not easy, and strongly depends on the task to be executed. A higher value of the active stiffness means a higher accuracy of the position control at the expense of higher interaction forces. Hence, if it is expected to meet some physical constraint in a particular direction, the end-effector stiffness in that direction should be made low to ensure low interaction forces. Conversely, along the directions where physical constraints are not expected, the end-effector stiffness should be made high so as to follow closely the desired position. This allows discrepancies between the desired and achievable positions due to the constraints imposed by the environment to be resolved without excessive contact forces and moments. It must be pointed out, however, that a selective stiffness behavior along different directions cannot be effectively assigned in practice on the basis of (7.6). This can easily be understood by using the classical definition of a mechanical stiffness for two bodies connected by a6-dof spring, in terms of the linear mapping between the infinitesimal twist displacement of the two bodies at an unloaded equilibrium and the elastic wrench. In the case of the active stiffness, the two bodies are, respectively, the end-effector, with the attached frame Σ e, and a virtual body, attached to the desired frame Σ d. Hence, from (7.6), the following mapping can be derived h e = A (ϕ e )K P A 1 (ϕ e )δx de, (7.7) in the case of an infinitesimal twist displacement δx de defined as ( ) ( ) ( ) δp δx de = de Δṗ = de ṗ dt = e dt, δθ de Δω de ω e where Δṗ de = ṗ d ṗ e is the time derivative of the position error Δp de = p d p e and Δω de = ω d ω e is the angular velocity error. Equation (7.7) shows that the actual stiffness matrix is A (ϕ e )K P A 1 (ϕ e ), which depends on the end-effector orientation through the vector ϕ e, so that, in practice, the selection of the stiffness parameters is quite difficult. This problem can be overcome by defining a geometrically consistent active stiffness, with the same structure and properties as ideal mechanical springs. Mechanical Springs Consider two elastically coupled rigid bodies A and B and two reference frames Σ a and Σ b, attached to A and B, respectively. Assuming that at equilibrium frames Σ a and Σ b coincide, the compliant behavior near the equilibrium can be described by the linear mapping ( ) h b b = Kδxb ab = K t K c δx b ab, (7.8) K c K o where h b b is the elastic wrench applied to body B, expressed in frame B, in the presence of an infinitesimal twist displacement δx b ab of frame Σ a with respect to frame Σ b, expressed in frame B. The elastic wrench and the infinitesimal twist displacement in (7.8) can also be expressed equivalently in frame Σ a, since Σ a and Σ b coincide at equilibrium. Therefore, h b b = ha b and δx b ab = δxa ab ; moreover, for the elastic wrench applied to body A, h a a = K tδx a ba = hb b being δxa ba = δxb ab. This property of the mapping (7.8) is known as port symmetry. In (7.8), K is the 6 6 symmetric positivesemidefinite stiffness matrix. The 3 3 matrices K t and K o, called respectively the translational stiffness and rotational stiffness, are also symmetric. It can be shown that, if the 3 3 matrix K c, called the coupling stiffness is symmetric, there is maximum decoupling between rotation and translation. In this case, the point corresponding to the coinciding origins of the frames Σ a and Σ b is called the center of stiffness. Similar definitions and results can be formulated for the case of the compliance matrix C = K 1. In particular, it is possible to define a center of compliance in the case that the offdiagonal blocks of the compliance matrix are symmetric. The center of stiffness and compliance do not necessarily coincide. There are special cases in which no coupling exists between translation and rotation, i. e., a relative transla- Part A 7.2

6 166 Part A Robotics Foundations Part A 7.2 tion of the bodies results in a wrench corresponding to a pure force along an axis through the center of stiffness; also, a relative rotation of the bodies results in a wrench that is equivalent to a pure torque about an axis through the centers of stiffness. In these cases, the center of stiffness and compliance coincide. Mechanical systems with completely decoupled behavior are, e.g., the remote center of compliance (RCC) devices. Since K t is symmetric, there exists a rotation matrix R t with respect to the frame Σ a = Σ b at equilibrium, such that K t = R t Γ t R t,andγ t is a diagonal matrix whose diagonal elements are the principal translational stiffnessess in the directions corresponding to the columns of the rotation matrix R t, known as the principal axes of translational stiffness. Analogously, K o can be expressed as K o = R o Γ o R o, where the diagonal elements of Γ o are the principal rotational stiffnesses about the axes corresponding to the columns of rotation matrix R o, known as the principal axes of rotational stiffness. Moreover, assuming that the origins of Σ a and Σ b at equilibrium coincide with the center of stiffness, the expression K c = R c Γ c R c can be found, where the diagonal elements of Γ c are the principal coupling stiffnesses along the directions corresponding to the columns of the rotation matrix R c, known as the principal axes of coupling stiffness. In sum, a 6 6 stiffness matrix can be specified, with respect to a frame with origin in the center of stiffness, in terms of the principal stiffness parameters and principal axes. Notice that the mechanical stiffness defined by (7.8) describes the behavior of an ideal 6-DOF spring which stores potential energy. The potential energy function of an ideal stiffness depends only on the relative position and orientation of the two attached bodies and is port symmetric. A physical 6-DOF spring has a predominant behavior similar to the ideal one, but nevertheless it always has parasitic effects causing energy dissipation. Geometrically Consistent Active Stiffness To achieve a geometrically consistent 6-DOF active stiffness, a suitable definition of the proportional control action in control law (7.3) is required. This control action can be interpreted as the elastic wrench applied to the end-effector, in the presence of a finite displacement of the desired frame Σ d with respect to the end-effector frame Σ e. Hence, the properties of the ideal mechanical stiffness for small displacements should be extended to the case of finite displacements. Moreover, to guarantee asymptotic stability in the sense of Lyapunov, a suitable potential elastic energy function must be defined. For simplicity, it is assumed that the coupling stiffness matrix is zero. Hence, the potential elastic energy can be computed as the sum of a translational potential energy and a rotational potential energy. The translational potential energy can be defined as with V t = 1 2 Δp de K Pt Δp de (7.9) K Pt = 1 2 R dk Pt R d R ek Pt R e, where K Pt is a 3 3 symmetric positive-definite matrix. The use of K Pt in lieu of K Pt in (7.9) guarantees that the potential energy is port symmetric also in the case of finite displacements. Matrices K Pt and K Pt coincide at equilibrium (i. e., when R d = R e ) and in the case of isotropic translational stiffness (i. e., when K Pt = k Pt I). The computation of the power V t yields V t = Δṗ e de f Δt e + Δωe de me Δt, where Δṗ e de is the time derivative of the position displacement Δp e de = R e (p d p e ), while Δω e de = R e (ω d ω e ). The vectors f Δ e and μe Δ are, respectively, the elastic force and moment applied to the end-effector in the presence of the finite position displacement Δp e de. These vectors have the following expressions when computed in the base frame with f Δt = K Pt Δp de m Δt = K Pt Δp de (7.10) K Pt = 1 2 S(Δp de)r d K Pt R d, where S( ) is the skew-symmetric operator performing the vector product. The vector h Δt = ( f Δt m Δt) is the elastic wrench applied to the end-effector in the presence of a finite position displacement Δp de and a null orientation displacement. The moment m Δt is null in the case of isotropic translational stiffness. To define the rotational potential energy, a suitable definition of the orientation displacement between the frames Σ d and Σ e has to be adopted. A possible choice is the vector part of the unit quaternion {η de,ɛ e de } that can be extracted from matrix R e d = R e R d. Hence, the orientation potential energy has the form V o = 2ɛ e de K Poɛ e de, (7.11) where K Po is a 3 3 symmetric positive-definite matrix. The function V o is port symmetric because ɛ e de = ɛd ed.

7 Force Control 7.2 Indirect Force Control 167 The computation of the power V o yields V o = Δω e de me Δo, where m Δo = K Po ɛ de, (7.12) with K Po = 2E (η de,ɛ de )R e K Po R e and E(η de,ɛ de ) = η de I S(ɛ de ). The above equations show that a finite orientation displacement ɛ de = R e ɛe de produces an elastic wrench h Δo = (0 m Δo ) which is equivalent to a pure moment. Hence, the total elastic wrench in the presence of a finite position and orientation displacement of the desired frame Σ d with respect to the end-effector frame Σ e can be defined in the base frame as h Δ = h Δt + h Δo. (7.13) where h Δt and h Δo are computed according to (7.10) and (7.12), respectively. Using (7.13) for the computation of the elastic wrench in the case of an infinitesimal twist displacement δx e de near the equilibrium, and discarding the high-order infinitesimal terms, yields the linear mapping ( ) h e e = K Pδx e de = K Pt 0 δx e de 0 K. (7.14) Po Therefore, K P represents the stiffness matrix of an ideal spring with respect to a frame Σ e (coinciding with Σ d at equilibrium) with the origin at the center of stiffness. Moreover, it can be shown, using definition (7.13), that the physical/geometrical meaning of the principal stiffnesses and of the principal axes for the matrices K Pt and K Po are preserved also in the case of large displacements. The above results imply that the active stiffness matrix K P can be set in a geometrically consistent way with respect to the task at hand. Notice that geometrical consistency can also be ensured with different definitions of the orientation error in the potential orientation energy (7.11), for example, any error based on the angle/axis representation of R d e can be adopted (the unit quaternion belongs to this category), or, more generally, homogeneous matrices or exponential coordinates (for the case of both position and orientation errors). Also, the XYZ Euler angles extracted from the matrix R d e could be used; however, in this case, it can be shown that the principal axes of rotational stiffness cannot be set arbitrarily but must coincide with those of the end-effector frame. Compliance control with a geometrically consistent active stiffness can be defined using the control law h c = h Δ K D v e + η(q), with h Δ in (7.13). The asymptotic stability about the equilibrium in the case h e = 0 can be proven using the Lyapunov function V = 1 2 v e Λ(q)v e + V t + V o, with V t and V o given in (7.9) and(7.11), respectively, whose time derivative along the trajectories of the closed-loop system, in case the frame Σ d is motionless, has the same expression as in (7.4). When h e = 0,adifferent asymptotically stable equilibrium can be found, corresponding to a nonnull displacement of the desired frame Σ d with respect to the end-effector frame Σ e. The new equilibrium is the solution of the equation h Δ = h e. Stiffness control allows to keep the interaction force and moment limited at the expense of the end-effector position and orientation error, with a proper choice of the stiffness matrix, without the need of a force/torque sensor. However, in the presence of disturbances (e.g., joint friction) which can be modeled as an equivalent end-effector wrench, the adoption of low values for the active stiffness may produce large deviations with respect to the desired end-effector position and orientation, also in the absence of interaction with the environment Impedance Control Stiffness control is designed to achieve a desired static behavior of the interaction. In fact, the dynamics of the controlled system depends on that of the robot manipulator, which is nonlinear and coupled. A more demanding objective may be that of achieving a desired dynamic behavior for the end-effector, e.g., that of a second-order mechanical system with six degrees of freedom, characterized by a given mass, damping, and stiffness, known as mechanical impedance. The starting point to pursue this goal may be the acceleration-resolved approach used for motion control, which is aimed at decoupling and linearizing the nonlinear robot dynamics at the acceleration level via an inverse dynamics control law. In the presence of interaction with the environment, the control law h c = Λ(q)α + Γ (q, q) q + h e (7.15) cast into the dynamic model (7.2) results in v e = α, (7.16) Part A 7.2

8 168 Part A Robotics Foundations Part A 7.2 p d, R d υ d υ d where α is a properly designed control input with the meaning of an acceleration referred to the base frame. Considering the identity v e = R e v e e + R e v e e, with ( ) R R e = e 0, 0 R e the choice gives α = R e α e + R e ve e (7.17) v e e = αe, (7.18) where the control input α e has the meaning of an acceleration referred to the end-effector frame Σ e. Hence, setting α e = K 1 M ( ve d + K DΔv e de + he Δ he e ), (7.19) the following expression can be found for the closedloop system K M Δ v e de + K DΔv e de + he Δ = he e, (7.20) where K M and K D are 6 6 symmetric and positivedefinite matrices, Δ v e de = ve d ve e, Δve de = ve d ve e, ve d and v e d are, respectively, the acceleration and the velocity of a desired frame Σ d and h e Δ is the elastic wrench (7.13); all the quantities are referred to the end-effector frame Σ e. The above equation describing the dynamic behavior of the controlled end-effector can be interpreted as a generalized mechanical impedance. The asymptotic stability of the equilibrium in the case h e = 0 can be proven by considering the Lyapunov function V = 1 2 Δve de K MΔv e de + V t + V o, (7.21) Impedance control α p e, R e υ e Fig. 7.1 Impedance control Inverse dynamics Direct kinematics τ Manipulator and environment h e q q where V t and V o are defined in (7.9)and(7.11), respectively, and whose time derivative along the trajectories of system (7.20) is the negative semidefinite function V = Δv e de K DΔv e de. When h e = 0, a different asymptotically stable equilibrium can be found, corresponding to a nonnull displacement of the desired frame Σ d with respect to the end-effector frame Σ e. The new equilibrium is the solution of the equation h e Δ = he. In case Σ d is constant, (7.20) has the meaning of atrue6-dof mechanical impedance if K M is chosen as ( K M = m I 0 0 M ), where m is a mass and M is a 3 3 inertia tensor, and K D is chosen as a block-diagonal matrix with 3 3 blocks. The physically equivalent system is a body of mass m with inertia tensor M with respect to a frame Σ e attached to the body, subject to an external wrench h e. This body is connected to a virtual body attached to frame Σ d through a 6-DOF ideal spring with stiffness matrix K P and is subject to viscous forces and moments with damping K D. The function V in (7.21) represents the total energy of the body: the sum of the kinetic and potential elastic energy. A block diagram of the resulting impedance control is sketched in Fig The impedance control computes the acceleration input as in (7.17) and(7.19) onthe basis of the position and orientation feedback as well as the force and moment measurements. Then, the inverse dynamics control law computes the torques for the joint actuators τ = J h c with h c in (7.15). This control scheme, in the absence of interaction, guarantees that the end-effector frame Σ e asymptotically follows the desired frame Σ d. In the presence of contact with the environment, a compliant dynamic behavior is imposed on the end-effector, according to the impedance (7.20), and the contact wrench is bounded at the expense of a finite position and orientation displacement between Σ d and Σ e. Differently from stiffness control, a force/torque sensor is required for the measurement of the contact force and moment. Implementation Issues The selection of good impedance parameters ensuring a satisfactory behavior is not an easy task. In fact, the dynamics of the closed-loop system is different in free space and during interaction. The control objectives are different as well, since motion tracking and disturbance

9 Force Control 7.2 Indirect Force Control 169 rejection must be ensured in free space, while, during the interaction, the main goal is achieving a suitable compliant dynamic behavior for the end-effector. Notice also that the dynamics of the controlled system during the interaction depends on the dynamics of the environment. To gain insight into these problems, assume that the interaction of the end-effector with the environment can be approximated by that derived from an ideal 6-DOF spring connecting end-effector frame Σ e to the environment frame Σ o. Therefore, according to (7.8), the elastic wrench exerted by the end-effector on the environment, in the presence of an infinitesimal twist displacement of Σ e with respect to Σ o, can be computed as h e e = Kδxe eo, (7.22) where Σ e and Σ o coincide at equilibrium and K is a stiffness matrix. The above model holds only in the presence of interaction, while the contact wrench is null when the end-effector moves in free space. The disturbances acting on the robot manipulator and the unmodeled dynamics (joint friction, modeling errors, etc.) may be taken into account by introducing an additive term on the right-hand side of the dynamic model of the robot manipulator (7.2), corresponding to an equivalent disturbance wrench acting on the end-effector. This term produces an additive acceleration disturbance γ e on the right-hand side of (7.18). Therefore, using the control law (7.19), the following closed-loop impedance equation can be found K M Δ v e de + K DΔv e de + he Δ = he e + K Mγ e. (7.23) The tuning procedure for the impedance parameters can be set up starting from the linearized model that can be computed from (7.23) in the case of infinitesimal displacements, i. e.: K M δẍ e de + K Dδẋ e de + (K P + K)δx e de = Kδx e do + K Mγ e, (7.24) where (7.22) and the equality δx e eo = δxe de + δxe do have been used. The above equation is valid both for constrained (K = 0) and for free motion (K = 0). It is evident that suitable dynamics of the position and orientation errors can be set by suitably choosing the matrix gains K M, K D,andK P.Thistaskiseasier under the hypothesis that all the matrices are diagonal, resulting in a decoupled behavior for the six components of the infinitesimal twist displacement. In this case, the transient behavior of each component can be set, e.g., by assigning the natural frequency and damping ratio with the relations k P + k ω n =, ζ = 1 k D k M 2 km (k P + k). Hence, if the gains are chosen so that a given natural frequency and damping ratio are ensured during the interaction (i. e., for k = 0), a smaller natural frequency with a higher damping ratio will be obtained when the end-effector moves in free space (i. e., for k = 0). As for the steady-state performance, the end-effector error for the generic component is k δx de = (k P + k) δx do + k M k P + k γ and the corresponding interaction force is h = k Pk k P + k δx do k Mk k P + k γ. The above relations show that, during interaction, the contact force can be made small at the expense of a large position error in steady state, as long as the active stiffness k P is set low with respect to the stiffness of the environment k, and vice versa. However, both the contact force and the position error also depend on the external disturbance γ ; in particular, the lower k P, the higher the influence of γ on both δx de and h. Moreover, a low active stiffness k P may result in a large position error also in the absence of interaction (i. e., when k = 0). Admittance Control A solution to this drawback can be devised by separating motion control from impedance control as follows. The motion control action is purposefully made stiff so as to enhance disturbance rejection but, rather than ensuring tracking of the desired end-effector position and orientation, it ensures tracking of a reference position and orientation resulting from the impedance control action. In other words, the desired position and orientation, together with the measured contact wrench, are input to the impedance equation which, via a suitable integration, generates the position and orientation to be used as a reference for the motion control. To implement this solution, it is worth introducing a reference frame other than the desired frame Σ d. This frame is referred to as the compliant frame Σ c,andis specified by the quantities p c, R c, v c,and v c that are computed from p d, R d, v d,and v d and the measured wrench h c, by integrating the equation K M Δ v c dc + K DΔv c dc + hc Δ = hc, (7.25) where h c Δ is the elastic wrench in the presence of a finite displacement between the desired frame Σ d and the Part A 7.2

10 170 Part A Robotics Foundations p d, R d υ d υ d Impedance control p c, R c υ c υ c Pos and orient control α Inverse dynamics τ Manipulator and environment h e q q p e, R e υ e Direct kinematics Fig. 7.2 Impedance control with inner motion control loop (admittance control) Part A 7.2 compliant frame Σ c. Then, a motion control strategy, based on inverse dynamics, is designed so that the endeffector frame Σ e is taken to coincide with the compliant frame Σ c. To guarantee the stability of the overall system, the bandwidth of the motion controller should be higher than the bandwidth of the impedance controller. A block diagram of the resulting scheme is sketched in Fig It is evident that, in the absence of interaction, the compliant frame Σ c coincides with the desired frame Σ d and the dynamics of the position and orientation error, as well as the disturbance rejection capabilities, depend only on the gains of the inner motion control loop. On the other hand, the dynamic behavior in the presence of interaction is imposed by the impedance gains (7.25). The control scheme of Fig. 7.2 is also known as admittance control because, in (7.25), the measured force (the input) is used to compute the motion of the compliant frame (the output), given the motion of the desired frame; a mapping with a force as input and a position or velocity as output corresponds to a mechanical admittance. Vice versa, (7.20), mapping the end-effector displacement (the input) from the desired motion trajectory into the contact wrench (the output), has the meaning of a mechanical impedance. Simplified Schemes The inverse dynamics control is model based and requires modification of current industrial robot controllers, which are usually equipped with independent PI joint velocity controllers with very high bandwidth. These controllers are able to decouple the robot dynamics to a large extent, especially in the case of slow motion, and to mitigate the effects of external forces on the manipulator motion if the environment is sufficiently compliant. Hence, the closed-loop dynamics of the controlled robot can be approximated by q = q r in joint space, or equivalently v e = v r (7.26) in the operational space, where q r and v r are the control signals for the inner velocity motion loop generated by a suitably designed outer control loop. These control signals are related by q r = J 1 (q)v r. The velocity v r, corresponding to a velocity-resolved control, can be computed as v e r = ve d + K 1 D (he Δ he e ), where the control input has been referred to the endeffector frame, K D is a 6 6 positive-definite matrix and h Δ is the elastic wrench (7.13) with stiffness matrix K P. The resulting closed-loop equation is K D Δv e de + he Δ = he e corresponding to a compliant behavior of the endeffector characterized by a damping K D and a stiffness K P. In the case K P = 0, the resulting scheme is known as damping control. Alternatively, an admittance-type control scheme can be adopted, where the motion of a compliant frame Σ c can be computed as the solution of the differential equation K D Δv c dc + hc Δ = hc e in terms of the position p c, orientation R c, and velocity twist v c, where the inputs are the motion variables of the desired frame Σ d and the contact wrench h c e. The motion variables of Σ c are then input to an inner position and velocity controller. In the case K D = 0, the resulting scheme is known as compliance control.

11 Force Control 7.3 Interaction Tasks Interaction Tasks Indirect force control does not require explicit knowledge of the environment, although to achieve a satisfactory dynamic behavior the control parameters have to be tuned for a particular task. On the other hand, a model of the interaction task is required for the synthesis of direct force control algorithms. An interaction task is characterized by complex contact situations between the manipulator and the environment. To guarantee proper task execution, it is necessary to have an analytical description of the interaction force and moment, which is very demanding from a modeling viewpoint. A real contact situation is a naturally distributed phenomenon in which the local characteristics of the contact surfaces as well as the global dynamics of the manipulator and environment are involved. In detail: the environment imposes kinematic constraints on the end-effector motion, due to one or more contacts of different type, and a reaction wrench arises when the end-effector tends to violate the constraints (e.g., the case of a robot sliding a rigid tool on a frictionless rigid surface); the end-effector, while being subject to kinematic constraints, may also exert a dynamic wrench on the environment, in the presence of environment dynamics (e.g., the case of a robot turning a crank, when the crank dynamics is relevant, or a robot pushing against a compliant surface); the contact wrench may depend on the structural compliance of the robot, due to the finite stiffness of the joints and links of the manipulator, as well as of the wrist force/torque sensor or of the tool (e.g., an end-effector mounted on an RCC device); local deformation of the contact surfaces may occur during the interaction, producing distributed contact areas (e.g., the case of a soft contact surface of the tool or of the environment); static and dynamic friction may occur in the case of non ideally smooth contact surfaces. The design of the interaction control and the performance analysis are usually carried out under simplifying assumptions. The following two cases are considered: 1. the robot and the environment are perfectly rigid and purely kinematics constraints are imposed by the environment, 2. the robot is perfectly rigid, all the compliance in the system is localized in the environment, and the contact wrench is approximated by a linear elastic model. In both cases, frictionless contact is assumed. It is obvious that these situations are only ideal. However, the robustness of the control should be able to cope with situations where some of the ideal assumptions are relaxed. In that case the control laws may be adapted to deal with nonideal characteristics Rigid Environment The kinematic constraints imposed by the environment can be represented by a set of equations that the variables describing the end-effector position and orientation must satisfy; since these variables depend on the joint variables through the direct kinematic equations, the constraint equations can also be expressed in the joint space as φ(q) = 0. (7.27) The vector φ is an m 1 function, with m < n, where n is the number of joints of the manipulator, assumed to be nonredundant; without loss of generality, the case n = 6 is considered. Constraints of the form (7.27), involving only the generalized coordinates of the system, are known as holonomic constraints. The case of time-varying constraints of the form φ(q, t) = 0 is not considered here but can be analyzed in a similar way. Moreover, only bilateral constraints expressed by equalities of the form (7.27) are of concern; this means that the end-effector always keeps contact with the environment. The analysis presented here is known as kinetostatic analysis. It is assumed that the vector (7.27) is twice differentiable and that its m components are linearly independent at least locally in a neighborhood of the operating point. Hence, differentiation of (7.27) yields J φ (q) q = 0, (7.28) where J φ (q) = φ/ q is the m 6 Jacobian of φ(q), known as the constraint Jacobian. By virtue of the above assumption, J φ (q) is of rank m at least locally in a neighborhood of the operating point. In the absence of friction, the generalized interaction forces are represented by a reaction wrench that tends to violate the constraints. This end-effector wrench produces reaction torques at the joints that can be computed Part A 7.3

12 172 Part A Robotics Foundations Part A 7.3 using the principle of virtual work as τ e = J φ (q)λ, where λ is an m 1 vector of Lagrange multipliers. The end-effector wrench corresponding to τ e can be computed as h e = J (q)τ e = S f (q)λ, (7.29) where S f = J (q)j φ (q). (7.30) From (7.29) it follows that h e belongs to the m- dimensional vector space spanned by the columns of the 6 m matrix S f. The inverse of the linear transformation (7.29) is computed as λ = S f (q)h e, (7.31) where S f denotes a weighted pseudoinverse of the matrix S f,i.e., S f = (S f WS f) 1 S f W (7.32) where W is a suitable weighting matrix. Notice that, while the range space of the matrix S f in (7.30) is uniquely defined by the geometry of the contact, the matrix S f itself is not unique; also, the constraint equations (7.27), the corresponding Jacobian J φ as well as the pseudoinverse S f and the vector λ are not uniquely defined. In general, the physical units of measure of the elements of λ are not homogeneous and the columns of the matrix S f,aswellasofthematrixs f, do not necessarily represent homogeneous entities. This may produce invariance problems in the transformation (7.31) ifh e represents a measured wrench that is subject to disturbances and, as a result, may have components outside the range space of S f. If a physical unit or a reference frame is changed, the matrix S f undergoes a transformation; however, the result of (7.31) with the transformed pseudoinverse in general depends on the adopted physical units or on the reference frame. The reason is that the pseudoinverse is the weighted least-squares solution of a minimization problem based on the norm of the vector h e S f (q)λ, and the invariance can be guaranteed only if a physically consistent norm of this vector is used. In the ideal case that h e is in the range space of S f, there is a unique solution for λ in (7.31), regardless of the weighting matrix, and hence the invariance problem does not appear. A possible solution consists of choosing S f so that its columns represent linearly independent wrenches. This implies that (7.29) givesh e as a linear combination of wrenches and λ is a dimensionless vector. A physically consistent norm on the wrench space can be defined based on the quadratic form h e K 1 h e, which has the meaning of an elastic energy if K is a positive-definite matrix corresponding to a stiffness. Hence, the choice W = K 1 can be made for the weighting matrix of the pseudoinverse. Notice that, for a given S f, the constraint Jacobian can be computed from (7.30) asj φ (q) = S f J(q); moreover, the constraint equations can be derived by integrating (7.28). Using (7.1) and(7.30), the equality (7.28) can be rewritten in the form J φ (q)j 1 (q)j(q) q = S f v e = 0, (7.33) which, by virtue of (7.29), is equivalent to h e v e = 0. (7.34) Equation (7.34) represents the kinetostatic relationship, known as reciprocity, between the ideal reaction wrench h e (belonging to the so-called force-controlled subspace) and the end-effector twist that obeys the constraints (belonging to the so-called velocity-controlled subspace). The concept of reciprocity, expressing the physical fact that, in the hypothesis of rigid and frictionless contact, the wrench does not cause any work against the twist, is often confused with the concept of orthogonality, which makes no sense in this case because twists and wrenches belong to different spaces. Equations (7.33) and(7.34) imply that the velocitycontrolled subspace is the reciprocal complement of the m-dimensional force-controlled subspace, identified by the range of matrix S f. Hence, the dimension of the velocity-controlled subspace is 6 m and a 6 (6 m) matrix S v can be defined, whose columns span the velocity-controlled subspace, i. e., v e = S v (q)ν, (7.35) where ν is a suitable (6 m) 1 vector. From (7.33) and (7.35) the following equality holds S f (q)s v(q) = 0 ; (7.36) moreover, the inverse of the linear transformation (7.35) can be computed as ν = S v (q)v e, (7.37) where S v denotes a suitable weighted pseudoinverse of the matrix S v, computed as in (7.32).

13 Force Control 7.3 Interaction Tasks 173 Notice that, as for the case of S f, although the range space of the matrix S v is uniquely defined, the choice of the matrix S v itself is not unique. Moreover, the columns of S v are not necessarily twists and the scalar ν may have nonhomogeneous physical dimensions. However, in order to avoid invariance problems analogous to that considered for the case of S f, it is convenient to select the columns of S v as twists so that the vector ν is dimensionless; moreover, the weighting matrix used to compute the pseudoinverse in (7.37) can be set as W = M,beingM a 6 6 inertia matrix; this corresponds to defining a norm in the space of twists based on the kinetic energy. It is worth observing that the transformation matrices of twists and wrenches, corresponding to a change of reference frame, are different; however, if twists are defined with angular velocity on top and translational velocity on bottom, then their transformation matrix is the same as for wrenches. The matrix S v may also have an interpretation in terms of Jacobians, as for S f in (7.30). Due to the presence of m independent holonomic constraints (7.27), the configuration of the robot in contact with the environment can be described in terms of a (6 m) 1 vector r of independent variables. From the implicit function theorem, this vector can be defined as r = ψ(q), (7.38) where ψ(q) isany(6 m) 1 twice-differentiable vector function such that the m components of φ(q) and the n m components of ψ(q) are linearly independent at least locally in a neighborhood of the operating point. This means that the mapping (7.38), together with the constraint (7.27), is locally invertible, with inverse defined as q = ρ(r), (7.39) where ρ(r) is a 6 1 twice-differentiable vector function. Equation (7.39) explicitly provides all the joint vectors which satisfy the constraint (7.27). Moreover, the joint velocity vectors that satisfy (7.28) can be computed as q = J ρ (r)ṙ, where J ρ (r) = ρ/ r is a 6 (6 m) full-rank Jacobian matrix. Therefore, the following equality holds J φ (q)j ρ (r) = 0, which can be interpreted as a reciprocity condition between the subspace of the reaction torques spanned by the columns of the matrix J φ and the subspace of the constrained joint velocities spanned by the columns of the matrix J ρ. Rewriting the above equation as J φ (q)j(q) 1 J(q)J ρ (r) = 0, and taking into account (7.30)and(7.36), the matrix S v can be expressed as S v = J(q)J ρ (r), (7.40) which, by virtue of (7.38) and(7.39), it can be equivalently expressed as a function of either q or r. The matrices S f and S v, and their pseudoinverse S f and S v are known as selection matrices. They play a fundamental role for the task specification, i. e., the specification of the desired end-effector motion and interaction forces and moments, as well as for the control synthesis Compliant Environment In many applications, the interaction wrench between the end-effector and a compliant environment can be approximated by an ideal elastic model of the form (7.22). However, since the stiffness matrix K is positive definite, this model describes a fully constrained case, when the environment deformation coincides with the infinitesimal twist displacement of the end-effector. In general, however, the end-effector motion is only partially constrained by the environment and this situation can be modeled by introducing a suitable positive-semidefinite stiffness matrix. The stiffness matrix describing the partially constrained interaction between the end-effector and the environment can be computed by modeling the environment as a couple of rigid bodies, S and O, connected through an ideal 6-DOF spring of compliance C = K 1. Body S is attached to a frame Σ s and is in contact with the end-effector; body O is attached to a frame Σ o which, at equilibrium, coincides with frame Σ s. The environment deformation about the equilibrium, in the presence of a wrench h s, is represented by the infinitesimal twist displacement δx so between frames Σ s and Σ o,which can be computed as δx so = Ch s. (7.41) All the quantities hereafter are referred to frame Σ s but the superscript s is omitted for brevity. For the considered contact situation, the end-effector twist does not completely belong to the ideal velocity subspace, corresponding to a rigid environment, because Part A 7.3

17 Force Control 7.4 Hybrid Force/Motion Control 177 a) b) y t y r y t y r f yt f yt θ Fig. 7.6a,b Estimation of orientation error: (a) velocitybased approach, (b) force-based approach Figure 7.6 reports an example of error between nominal and measured motion and force variables, typical of f xt υ xt x t θ υ yt x t υ x r x r a two-dimensional contour-following task. The orientation of the contact normal changes if the environment is notplanar. Hence an angular error θ appears between the nominal contact normal, aligned to the y t -axis of the task frame (the frame with axes x t and y t ), and the real contact normal, aligned to the y r -axis of the real task frame (the frame with axes x r and y r ). This angle can be estimated with either velocity or force measurements only. Velocity-based approach: the executed linear velocity v, which is tangent to the real contour (aligned to the x r -axis), does not completely lie along the x t -axis, but has a small component v yt along the y t -axis. The orientation error θ can then be approximated by θ = tan 1 (v yt /v xt ). Force-based approach: the measured (ideal) contact force f does not completely lie along the nominal normal direction, aligned to the y t -axis, but has a small component f xt along the x t -axis. The orientation error θ can then be approximated by θ = tan 1 ( f xt / f yt ). The velocity-based approach is disturbed by mechanical compliance in the system; the force-based approach is disturbed by contact friction. 7.4 Hybrid Force/Motion Control The aim of hybrid force/motion control is to split up simultaneous control of both end-effector motion and contact forces into two separate decoupled subproblems. In the following, the main control approaches in the hybrid framework are presented for the cases of both rigid and compliant environments Acceleration-Resolved Approach As for the case of motion control, the accelerationresolved approach is aimed at decoupling and linearizing the nonlinear robot dynamics at the acceleration level, via an inverse dynamics control law. In the presence of interaction with the environment, a complete decoupling between the force- and velocity-controlled subspaces is sought. The basic idea is that of designing a model-based inner control loop to compensate for the nonlinear dynamics of the robot manipulator and decouple the force and velocity subspaces; hence an outer control loop is designed to ensure disturbance rejection and tracking of the desired end-effector force and motion. Rigid Environment In the case of a rigid environment, the external wrench can be written in the form h e = S f λ. The force multiplier λ can be eliminated from (7.2) by solving (7.2) for v e and substituting it into the time derivative of the last equality (7.33). This yields: { λ = Λ f (q) S f Λ 1 (q) [ h c μ(q, q) ] } + Ṡ f v e, (7.49) where Λ f (q) = (S f Λ 1 S f ) 1 and μ(q, q) = Γ q + η. Therefore the constraint dynamics can be rewritten in the form Λ(q) v e + S f Λ f (q)ṡ f v e = P(q) [ h c μ(q, q) ], (7.50) where P = I S f Λ f S f Λ 1. Notice that PS f = 0, hence the 6 6 matrix P is a projection matrix that filters out all the end-effector wrenches lying in the range of S f. These correspond to wrenches that tend to violate the constraints. Part A 7.4

18 178 Part A Robotics Foundations Part A 7.4 Equation (7.49) reveals that the force multiplier vector λ instantaneously depends also on the applied input wrench h c. Hence, by suitably choosing h c, it is possible to directly control the m independent components of the end-effector wrench that tend to violate the constraints; these components can be computed from the m force multipliers through (7.29). On the other hand, (7.50) represents a set of six second-order differential equations whose solution, if initialized on the constraints, automatically satisfy (7.27) at all times. The reduced-order dynamics of the constrained system is described by 6 m second-order equations that are obtained by premultiplying both sides of (7.50) by the matrix S v and substituting the acceleration v e with v e = S v ν + Ṡ v ν. The resulting equations are Λ v (q) ν = S [ v hc μ(q, q) Λ(q)Ṡ v ν ], (7.51) where Λ v = S v ΛS v and the identities (7.36)andS v P = S v have been used. Moreover, expression (7.49) can be rewritten as λ = Λ f (q)s f Λ 1 (q) [ h c μ(q, q) Λ(q)Ṡ v ν ], where the identity Ṡ f S v = S f Ṡ v has been exploited. An inverse-dynamics inner control loop can be designed by choosing the control wrench h c as h c = Λ(q)S v α v + S f f λ + μ(q, q) + Λ(q)Ṡ v ν, (7.52) where α v and f λ are properly designed control inputs. Substituting (7.52)into(7.51)and(7.49) yields ν = α ν λ = f λ, showing that control law (7.52) allows complete decoupling between the force- and velocity-controlled subspaces. It is worth noticing that, for the implementation of the control law (7.52), the constraint (7.27) as well as (7.38) defining the vector of the configuration variables for the constrained system are not required, provided that the matrices S f and S v are known or estimated online. In these cases, the task can easily be assigned by specifying a desired force, in terms of the vector λ d (t), and a desired velocity, in terms of the vector ν d (t); moreover, a force/velocity control is implemented. The desired force λ d (t) can be achieved by setting f λ = λ d (t), (7.53) but this choice is very sensitive to disturbance forces, since it contains no force feedback. Alternative choices are or f λ = λ d (t) + K Pλ [ λd (t) λ(t) ], (7.54) f λ = λ d (t) + K Iλ t 0 [ λd (τ) λ(τ) ] dτ, (7.55) where K Pλ and K Iλ are suitable positive-definite matrix gains. The proportional feedback is able to reduce the force error due to disturbance forces, while the integral action is able to compensate for constant bias disturbances. The implementation of force feedback requires the computation of the force multiplier λ from the measurement of the end-effector wrench h e, which can be achieved by using (7.31). Velocity control is achieved by setting α ν = ν d (t) + K Pν [ νd (t) ν(t) ] + K Iν t 0 [ νd (τ) ν(τ) ] dτ, (7.56) where K Pν and K Iν are suitable matrix gains. It is straightforward to show that asymptotic tracking of ν d (t) and ν d (t) is ensured with exponential convergence for any choice of positive-definite matrices K Pν and K Iν. The computation of the vector ν from the available measurements can be achieved using (7.37), where the end-effector twist is computed from joint position and velocity measurements through (7.1). Equations (7.54) or(7.55) and(7.56) represent the outer control loop ensuring force/velocity control and disturbance rejection. When equations (7.27) and(7.38) are known, the matrices S f and S v can be computed according to (7.30) and (7.40) andaforce/position control can be designed by specifying a desired force λ d (t) and a desired position r d (t). Force control can be designed as above, while position control can be achieved by setting α ν = r d (t) + K Dr [ṙd (t) ν(t) ] + K Pr [ rd (t) r(t) ]. Asymptotic tracking of r d (t), ṙ d (t) and r d (t) is ensured with exponential convergence for any choice of positivedefinite matrices K Dr and K Pr. The vector r required for position feedback can be computed from joint position measurements via (7.38).

19 Force Control 7.4 Hybrid Force/Motion Control 179 Compliant Environment In the case of a compliant environment, according to the decomposition (7.42) of the end-effector displacement, the end-effector twist can be decomposed as v e = S v ν + C S f λ, (7.57) where the first term is a twist of freedom, the second term is a twist of constraint, the vector ν is defined as in (7.40), and C is defined in (7.48). Assuming a constant contact geometry and compliance, i. e., Ṡ v = 0, Ċ = 0, andṡ f = 0, a similar decomposition holds in terms of acceleration v e = S v ν + C S f λ. (7.58) An inverse-dynamics control law (7.15) can be adopted, resulting in the closed loop (7.16), where α is a properly designed control input. In view of the acceleration decomposition (7.58), the choice α = S v α ν + C S f f λ (7.59) allows decoupling of the force control from velocity control. In fact, substituting (7.58)and(7.59)into(7.16) and premultiplying both sides of the resulting equation once by S v and once by S f, the following decoupled equations can be derived ν = α ν (7.60) λ = f λ. (7.61) Hence, by choosing α ν according to (7.56) asforthe rigid environment case, asymptotic tracking of a desired velocity ν d (t) and acceleration ν d (t) is ensured, with exponential convergence. The control input f λ can be chosen as f λ = λ d (t) + K Dλ [ λ d (t) λ(t) ] [ + K Pλ λd (t) λ(t) ], (7.62) ensuring asymptotic tracking of a desired force trajectory ( λ d (t), λ d (t), λ d (t) ) with exponential convergence for any choice of positive-definite matrices K Dλ and K Pλ. Differently from the rigid environment case, feedback of λ is required for the implementation of the force control law (7.62). This quantity could be computed from end-effector wrench measurements h e as λ = S f ḣe. However, since the wrench measurement signal is often noisy, the feedback of λ is often replaced by λ = S f K J(q) q, (7.63) where joint velocities are measured using tachometers or computed from joint positions via numerical differentiation and K is the positive-semidefinite stiffness matrix (7.47) describing the partially constrained interaction. For the computation of (7.63), only knowledge (or an estimate) of K is required, and not that of the full stiffness matrix K. Also, the implementation of the control law (7.59) requires knowledge (or an estimate) of the compliance matrix C of the partially constrained interaction and not that of the full compliance matrix C. If the contact geometry is known, but only an estimate of the stiffness/compliance of the environment is available, the control law (7.59), with (7.62), may still guarantee the convergence of the force error, if a constant desired force λ d is assigned. In this case, the control law (7.59)hastheform α = S v α ν + Ĉ S f f λ, where Ĉ = (I P v )Ĉ and Ĉ is an estimate of the compliance matrix. Hence, (7.60) still holds, while, in lieu of (7.61), the following equality can be found λ = L f f λ where L f = (S f CS f) 1 S f ĈS f is a nonsingular matrix. Thus, the force- and velocity-controlled subspaces remain decoupled and the velocity control law (7.56) does not need to be modified. On the other hand, if the feedback of the time derivative of λ is computed using (7.63), only an estimate ˆλ can be obtained. Using (7.63), (7.57) and (7.46), the following equality can be found ˆλ = L 1 f λ. Therefore, computing the force control law f λ as in (7.62) with a constant λ d, ˆλ in lieu of λ and K Dλ = K Dλ I, the dynamics of the closed-loop system is λ + K Dλ λ + L f K Pλ λ = L f K Pλ λ d, showing that exponential asymptotic stability of the equilibrium λ = λ d can be ensured, also in the presence of the uncertain matrix L f, with a suitable choice of the gains K Dλ and K Pλ Passivity-Based Approach The passivity-based approach exploits the passivity properties of the dynamic model of the manipulator, which hold also for the constrained dynamic model (7.2). It can be easily shown that the choice of the matrix Part A 7.4

20 180 Part A Robotics Foundations Part A 7.4 C(q, q) that guarantees the skew symmetry of the matrix Ḣ(q) 2C(q, q) in the joint space, also makes the matrix Λ(q) 2Γ (q, q) skew symmetric. This fundamental property of Lagrangian systems is at the base of passivity-based control algorithms. Rigid Environment The control wrench h c can be chosen as h c = Λ(q)S v ν r + Γ (q, q)ν r + (S v ) K ν (ν r ν) + η(q) + S f f λ, (7.64) where Γ (q, q) = Γ S v + ΛṠ v, K ν is a suitable symmetric and positive-definite matrix, and ν r and f λ are properly designed control inputs. Substituting (7.64)into(7.2) yields Λ(q)S v ṡ ν + Γ (q, q)s ν + (S v ) K ν s ν + S f ( f λ λ) = 0 (7.65) with ṡ ν = ν r ν and s ν = ν r ν, showing that the closed-loop system remains nonlinear and coupled. Premultiplying both sides of (7.65) by the matrix S v, the following expression for the reduced-order dynamics is achieved Λ v (q)ṡ ν + Γ v (q, q)s ν + K ν s ν = 0, (7.66) with Γ v = S v Γ (q, q)s v + S v Λ(q)Ṡ v ; it can easily be shown that the skew symmetry of the matrix Λ(q) 2Γ (q, q) implies that the matrix Λ v (q) 2Γ v (q, q) is skew symmetric as well. On the other hand, premultiplying both sides of (7.65) by the matrix S f Λ 1 (q), the following expression for the force dynamics can be found f λ λ = Λ f (q)s f Λ 1 (q) [ Γ (q, q) (S ] v ) K ν sν, (7.67) showing that the force multiplier λ instantaneously depends on the control input f λ but also on the error s ν in the velocity-controlled subspace. The asymptotic stability of the reduced-order system (7.66) can be ensured with the choices ν r = ν d + αδν, (7.68) ν r = ν d + αδx ν, (7.69) where α is a positive gain, ν d and ν d are the desired acceleration and velocity, respectively, Δν = ν d ν,and t Δx ν = Δν(τ)dτ. 0 The stability proof is based on the positive-definite Lyapunov function V = 1 2 s ν Λ v(q)s ν + αδx ν K νδx ν, whose time derivative along the trajectories of (7.66), V = Δν K ν Δν α 2 Δx ν K νδx ν, is a definite semi-negative function. Hence, Δν = 0, Δx ν = 0,ands ν = 0 asymptotically. Therefore, tracking of the desired velocity ν d (t) is ensured. Moreover, the right-hand side of (7.67) remains bounded and vanishes asymptotically. Hence, tracking of the desired force λ d (t) can be ensured by setting f λ as for the accelerationresolved approach, according to the choices (7.53), (7.54), or (7.55). Notice that position control can be achieved if a desired position r d (t) is assigned for the vector r in (7.38), provided that the matrices S f and S v are computed according to (7.30) and(7.40), and the vectors ν d = r d, ν d = ṙ d,andδx ν = r d r are used in (7.68)and(7.69). Compliant Environment The control wrench h c can be chosen as h c = Λ(q) v r + Γ (q, q)v r + K s (v r v e ) + h e + η(q), (7.70) where K s is a suitable symmetric positive-definite matrix while v r and its time derivative v r are chosen as v r = v d + αδx, v r = v d + αδv, where α is the positive gain, v d and its time derivative v d are properly designed control inputs, Δv = v d v e, and t Δx = Δvdτ. 0 Substituting (7.70)into(7.2) yields Λ(q)ṡ + Γ (q, q)s + K s s = 0 (7.71) with ṡ = v r v e and s = v r v e. The asymptotic stability of system (7.71) can be ensured by setting: v d = S v ν d + C S f λ d, where ν d (t) is a desired velocity trajectory and λ d (t) is the desired force trajectory. The stability proof is based on the positive-definite Lyapunov function V = 1 2 s Λ(q)s + αδx K s Δx,

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