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2 norm of the tracking error over the task space constraint. This function is different from the logarithmic type barrier function of [2], which was formulated in terms of the tracking error only, and necessitated a mapping of the task space constraint to a dynamic error space constraint. Another difference is that [2] only handles box constraints, while the BLF proposed here allows the handling of more general constraints such as ellipsoids. Compared with box constraints, ellipsoidal constraints are harder to deal with, since the constraint boundary is a nonlinear curve or surface in task space, and cannot be handled in a decoupled manner along each dimension of the task space as in the case of box constraints. Our new barrier function simplifies the control design and enables the direct handling of ellipsoidal task space constraints without prior mapping to the error space, which can be a complicated process and can lead to a conservative design. Under a unified framework of BLFbased Lyapunov control synthesis and stability analysis, we show that the end-effector is able to track a desired trajectory while never leaving the ellipsoidal constrained region. II. PROBLEM FORMULATION Throughout this paper, we denote by R the set of real numbers, and the Euclidean vector norm in R m. A. Robot Dynamics Consider a robot manipulator described by: M(q) q + C(q, q) q + G(q) + F( q) = τ + τ e (t) () where M(q) R n n is a symmetric positive definite matrix, C(q, q) q R n the Coriolis and centrifugal forces, G(q) R n the gravitational forces, F( q) R n the frictional forces, q R n the robot joint position, τ R n the input torque, and τ e R n a bounded interaction torque from the environment (or human) satisfying τ e (t) τ e, t, with τ e a constant. The terms M(q), C(q, q), F( q) and G(q) contain uncertain dynamic parameters. To track a desired trajectory in task space, the joint space dynamics () are transformed into task space dynamics [3]: M x ẍ + C x ẋ + G x + F x = f + f e (t) (2) via the forward kinematics and the Jacobian: x = Ω(q), ẋ = Ω q =: J(q) q (3) q where x = [x,,...,x m ] T is a vector of task variables, and the coefficient matrices are defined as M x = J T MJ, G x = J T G, F x = J T F C x = J T (C MJ J)J, f = J T τ, f e = J T τ e (4) Note that the external force f e is bounded by a constant f e, i.e. f e (t) f e, t. For simplicity, we consider only non-redundant (m = n) non-singular manipulators with known Jacobian J in this paper. Henceforth, we consider x R n. Furthermore, we consider only positioning of the end-effector (i.e. n 3), although our method can be easily modified to control end-effector orientation too. The following properties hold [3]. Property : The inertia matrix M x is symmetric positive definite. Property 2: The matrix Ṁx 2C x is skew symmetric. Property 3: The left-hand-side expression of (2) can be linearly parameterized in terms of the robot system parameters as follows: M x φ + C x φ 2 + G x + F x = ψ(φ,φ 2,x,ẋ)θ (5) for any φ,φ 2 R m, where θ R l are constant parameters and ψ R l is a known regressor function. B. Control Objective The control objective is to ensure that the end-effector position x = [x,...,x n ] T tracks a desired trajectory x d = [x d,...,x dn ] T while always satisfying position constraints imposed on the end-effector. Specifically, the end-effector is required to satisfy position constraints, which involve always staying in the interior of an ellipsoidal region x T Sx =, where S is a positive definite symmetric matrix. Thus, the position constraint set can be described by x Ω x, where Ω x = {x R n : x T Sx < } (6) In other words, the controller needs to ensure that x(t) Ω x t >, given that x() Ω x. Assumption : The desired trajectory x d (t) Ω x t, and there exist positive constants Y and Y 2 such that ẋ d (t) Y and ẍ d (t) Y 2, t. III. CONTROL SYNTHESIS USING A BARRIER LYAPUNOV FUNCTION Denote the error variables as: z = [z,...,z n ] T = x x d w = [w,...,w n ] T = ẋ α (7) where α = [α,...,α n ] T is a stabilizing function to be designed shortly. Then, we define the following constraint functions: (x) = x T Sx (8) For system (2), consider the Barrier Lyapunov Function (BLF) candidate V (z,x,w,q, θ) = V z (z,x) + V w (w,q) + V θ ( θ) (9) z T z V z (z,x) = () 2 (x) V w (w,q) = 2 wt M x w () V θ ( θ) = 2 θ T Γ θ (2) where σ = [σ,...,σ n ] T, Γ is a positive definite matrix, θ = ˆθ θ is the error between θ and its estimate ˆθ, and () is 68

3 a quotient of mixed variables z and x. It can be shown that V is positive definite and continuously differentiable in the set x Ω x. We employ adaptive backstepping procedure [4] to design the controller. For the robot manipulator (), the procedure consists of 2 steps. Step First, we design the stabilizing function. To this end, we obtain the partial derivatives of V z (z,x) as follows: V z (z,x) z V z (z,x) = z T (x) = zt z 2g 2 s(x) (x) Thus, the time-derivative of V z in () is given by V z = where = z T zt z 2g 2 s(x) (3) (x) (w + α) (x) (w + α ẋ d) z T (x) [h(z,x)(w + α) ẋ d] (4) h(z, x) = ( (x) = xt Sx d x T Sx ) z 2 (x) We design the stabilizing function α as follows: α(z,x,ẋ d ) = (5) h(z,x) ( K zz + ẋ d ) (6) where K z is a positive definite matrix. By substituting (6) into (4), we obtain: V z = zt K z z + h(z,x)zt w (7) The following lemma shows that h(z,x) is bounded away from in the set x,x d Ω x, which is important for ensuring that α(z,x,ẋ d ) is bounded for x,x d Ω x and bounded ẋ d. Lemma : The term h(z, x), defined in (5), satisfies h(z,x) > in the set x,x d Ω x. Proof: By completion of squares, we obtain the inequality: 2x T Sx d x T Sx + x T d Sx d (8) Furthermore, in the set x,x d Ω x, we have x T Sx < and x T d Sx d <. Then, it is clear, from (8), that x T Sx d < (9) Finally, from (5), it follows that h(z,x) > in the set x,x d Ω x. Step 2 In this step, we design the adaptive control torque and the adaptation law. Similar to Step, we obtain the timederivative of V w from (2) and () as: V w = w T ( C x ẋ G x F x + f + f e M x α) + 2 wt Ṁw (2) Substituting ẋ = w + α and using Property 2, we eliminate the last term on the right hand side and obtain V w = w T ( C x α G x F x + f + f e M x α) (2) Then, based on Property 3, we can rewrite V w such that the robot dynamics are linearly parameterized: V w = w T ( ψ( α,α,x,ẋ)θ + f + f e ) (22) where ψ is a known regressor. Then, we obtain the timederivative of the Lyapunov function candidate (9) as V = zt K z z + w T ( ψ( α,α,x,ẋ)θ + f + f e ) + θ T Γ ˆθ (23) Now, we design the control as f = ψ( α,α,x,ẋ)ˆθ K w w h(z,x)z (24) τ = J T (q)f (25) where K w > I/2 is a positive definite matrix. For robust adaptation of the parameter estimate ˆθ, we employ the following adaptation law: ˆθ = Γ( ψ( α,α,x,ẋ)w µˆθ) (26) where µ is a positive constant. The leakage term provides robustness against unmodelled disturbances and noisy force measurements. Substituting (24) and (26) into (23), it can be shown that V zt K z z w T (K w 2 I)w + 2 f 2 e µ 2 θ 2 + µ 2 θ 2 ρv + c (27) in the set x Ω x, where ρ = { min 2λ min (K z ), 2λ min(k w 2 I) } µ, λ max (M x ) λ max (Γ ) (28) c = 2 f 2 e µ 2 θ 2 + µ 2 θ 2 (29) The closed loop error system is given by: ż = h(z,x) ( K zz + ẋ d ) + w ẋ d ( ẇ = Mx ψ( α,α,x,ẋ) θ K w w h(z,x)z ) θ = Γ( ψ( α,α,x,ẋ)w µ( θ + θ)) (3) which can be rewritten in the form ξ = h(t,ξ) (3) where ξ = [z,w, θ] T, h(t,ξ) is piecewise continuous in t and locally Lipschitz in ξ, uniformly in t, in the set ξ Ψ = {ξ R 2n+l : > } (32) 69

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