Motion Planning for Dnamic Variable Inertia Mechanical Sstems with Non-holonomic Constraints Elie A. Shammas, Howie Choset, and Alfred A. Rizzi Carnegie Mellon Universit, The Robotics Institute Pittsburgh, PA 53, U.S.A. eshammas@andrew.cmu.edu, choset@cs.cmu.edu, arizzi@cs.cmu.edu Abstract: In this paper, we address a particular flavor of the motion planning problem, that is, the gait generation problem for underactuated variable inertia mechanical sstems. Additionall, we analze a rather general tpe of mechanical sstems which we refer to as mied sstems. What is unique about this tpe of mechanical sstem is that both non-holonomic velocit constraints as well as instantaneous conservation of the generalized momentum variables defined along the allowable motion direction completel specif the sstems velocit. B analzing this general tpe of mechanical sstems, we la the grounds for a general and intuitive analsis of the gait generation problem. Through our approach, we provide a novel framework not onl for classifing different tpes of mechanical sstems, but also for identifing a partition on the space of allowable gaits. B appling our techniques to mied sstems which according to our classification are the most general tpe of mechanical sstems, we verif the generalit and applicabilit of our approach. Moreover, mied sstems ield the richest famil of allowable gaits, hence, superseding the gait generation problem for other simpler tpes of mechanical sstems. Finall, we appl our analsis to a novel mechanical sstem, the variable inertia snakeboard, which is a generalization of the original snakeboard that was previousl studied in the literature. Introduction It is straight forward to analze the motion of a mechanical sstem due to a particular set of inputs or generalized forces. In fact, this is done b solving a set of second order non-linear differential equations of motion, usuall referred to as the Euler-Lagrange equations of motion. In most cases, the solution of this set of differential equations is numericall computed, since in general the do not ield a closed form solution. The gait generation problem involves solving the reverse problem, that is, finding a set of inputs that will cause a desired behavior of motion of the mechanical sstem. Given, the highl non-linear nature of the governing equations of motion, one can appreciate the
Elie A. Shammas, Howie Choset, and Alfred A. Rizzi difficult and compleit of the gait generation problem for mechanical sstems. To further complicate the gait generation problem, we gear our analsis toward underactuated mechanical sstems, that is, sstems that do not have as man actuators as the number of degrees of freedom. Finall, we also seek to verif the generalit of our gait generation techniques b analzing, according to our classification, the most general tpe of mechanical sstems and b ensuring that the sstems do not benefit from a fied inertia propert which considerabl simplifies the epressions of the equations of motion. In our prior work, we unified the gait generation approach for two seemingl different families of mechanical sstems, principall kinematic and purel mechanical sstems in [] and [4], respectivel. Even though, these two families of sstems belong to the opposite ends of a spectrum where at one end are purel mechanical sstems whose motion is governed solel b the conservation of momentum while at the other end are principall kinematic sstems whose motion is governed solel b the eistence of a set of independent nonholonomic constraints that full constrain the sstems velocit, we devised a rather simple gait evaluation tool which is equall applicable to both tpes of sstems. The fact that we proved that momentum, is null for the case of purel mechanical sstems and non-eistent for the case of the principall kinematic sstems allowed us to intuitivel generate geometric gaits for both sstems. Nonetheless, for mied sstem, in general we can not neglect the sstem s momentum as it could be the dominating contributing factor of motion for certain gaits. At the core of our dnamic gait analsis is a deep understanding of a generalized notion of the sstems momentum and its time evolution. We attain this goal through a twofold reduction or simplification of the equations of motion. First, as shown in the prior work, we utilize the smmetr in the laws of phsics to represents the evolution of the momentum as a first order differential equation. Additionall, we devised a second reduction step to further simplif the epression of this evolution equation so that we can use to intuitivel generate dnamic gaits. In this paper, we generate gaits for a novel mechanical sstem, the variable inertia snakeboard, shown in Fig. (a). This sstem is a generalization of the original snakeboard, (Fig. (b)), which was etensivel studied in the literature, [, 8], and which we analzed in [3]. Both snakeboards belong to the mied tpe sstems, that is, the non-holonomic constraints do not full span the fiber space. Thus, the generalized non-holonomic momentum must be instantaneousl conserved along certain directions which for the above snakeboards are rotations about the wheel aes intersections. However, the inertia of the original snakeboard is independent of the base variables which greatl simplifies the gait generation analsis. Thus, we consider the variable inertia Changing the base variables, rotor and wheel aes angles, in Fig. (b) will neither change the position of the sstem s center of mass, nor change its inertia about that point.
Motion Planning for Variable Inertia Mechanical Sstems 3 R ξ Bod Frame (,) α L (a) ξ R α θ α ξ L α α ξ L θ (,) (b) Fig.. A schematic of the variable inertia snakeboard in (a) and original snakeboard in (b) depicting their configuration variables. snakeboard, which as its name suggests, has a non-constant inertia, to verif the generalit and applicabilit of our gait analsis techniques. Utilizing our gait generation techniques we design curves in the actuated base space which represents the internal degrees of freedom of the robot to translate and rotate the variable inertia snakeboard in the plane. In other words, we will generate gaits b using the actuated base variables to control the un-actuated variables of the fiber space which denote the position of the sstem with respect to a fied inertial frame. Thus, our goal is to design cclic curves in the base space, which after a complete ccle, produce a desired motion along a specified fiber direction, hence, effectivel moving the robot to a new position. We start b presenting the following related prior approaches. Sinusoidal inputs: Ostrowski et. al. epressed the dnamics of a mechanical sstem in bod coordinates and were able to represent it as an affine non-linear control sstem. Then b taking recourse to control theor, the were able to design sinusoidal gaits and specif the gait frequencies. Nonetheless, the gait amplitudes were empiricall derived [9]. Ostrowski et. al. used their gait generation analsis to generate gaits for the original snakeboard (Fig. (b)) [8]. Moreover, Chitta et. al. developed several unconventional locomoting robots, such as the robo-trikke and the rollerblader, [4, ], then used Ostrowski s techniques to generate sinusoidal gaits for these novel locomoting robots. Prior work related to Ostrowski s can also be found in [, 7, 5]. Kinematic reduction: The work done b Bullo et. al. in [3] on kinematic reduction of simple mechanical sstems is closel related to our work. The define a kinematic reduction for simple mechanical sstems, or in other words, reduce the dnamics of a sstem so that it can be represented as a kinematic sstem. Then the stud the controllabilit of these reduced sstems and for certain eamples, the were able to generate gaits for these sstems. In fact, Bullo et. al. have designed gaits for the original snakeboard (Fig. (b)) which we analzed in [3]. In this paper, we introduce one tpe of gait, a purel kinematic gait, which is structurall similar to gaits proposed b Bullo et. al. in []; however, we have a different wa of generating these gaits.
4 Elie A. Shammas, Howie Choset, and Alfred A. Rizzi Background Material Here we present a rather abbreviated introduction to Lagrangian mechanics, introduce mied sstems, and finall we present several mechanics of locomotion results which we shall utilize to generate gaits for mied sstems. Lagrangian mechanics: The n-dimensional configuration space of a mechanical sstem, usuall denoted b Q, is a trivial principal fiber bundle; that is, Q = G M where G is the fiber space which has a Lie group structure and M is the base space. In this paper we assume the Lagrangian of a mechanical sstem to be its kinetic energ. Moreover, we assume that the non-holonomic constraints that are acting on the mechanical sstem can be written in a Pfaffian from, ω(q) q =, where ω(q) is a k n matri and q represents an element in the tangent space of the configuration manifold Q. Associated with the Lie group structure of the fiber space, G, we can define the action, Φ g, and the lifted action, T g Φ g, which act on the entire configuration manifold, Q, and tangent bundle, T Q, respectivel. Since we can verif that both the Lagrangian and non-holonomic constraints are invariant with respect to these action, we can epress the sstem s dnamics at the Lie group identit as was shown in [5]. In other words, we eliminate the dependence on the placement of the inertial frame. This invariance allows us to compute the reduced Lagrangian, l(ξ,r,ṙ), which according to [8] will have the form shown in () and the reduced non-holonomic constraints shown in () as we demonstrated in []. «T ξ M ṙ «ξ = ṙ «l(ξ, r, ṙ) = «ξ = ` ω ṙ ξ (r) ω ξ r(r) ṙ ω(r) «T ξ ṙ I(r) I(r)A(r) A T (r)i T (r) m(r) ««ξ ṙ () = () Here M is the reduced mass matri, A(r) is the local form of the mechanical connection, I(r) is the local form of the locked inertia tensor, that is, I(r) = I(e,r) 3, and m(r) is a matri depending onl on base variables. Finall, recall that ξ is an element of the Lie algebra and is given b ξ = T g L g ġ, where T g L g is the lifted action acting on a tangent space element ġ. Mied sstems: Such sstems are a general tpe of dnamic mechanical sstem that are subject to a set of non-holonomic constraints which are invariant with respect to the Lie group action. Hence, a mechanical sstem whose configuration space has a trivial principal fiber structure, Q = G M, and is subjected to k non-holonomic constraints, ω(q) q =, is said to be mied if the number of constraints acting on it is less than the dimension of Note that the elements of the tangent space at the fiber space identit form a Lie algebra which is usuall denoted b g. 3 e is the Lie group identit element.
Motion Planning for Variable Inertia Mechanical Sstems 5 the sstem s fiber space ( < k < l), the constraints are linearl independent (det(ω(q)) ), and the constraints are invariant with respect to the Lie group actions (ω(q) q = ω(φ g (q)) T g Φ g ( q) = ). Mechanics of locomotion: Now we borrow some well-known results from the mechanics of locomotion, [6], upon which we shall build our own gait generation techniques. For a mied sstem, according to [8] the sstem s configuration velocit epressed in bod coordinates, ξ, is given b the reconstruction equation shown in (3), where A(r) is an l m matri denoting the local form of the mied non-holonomic connection, Γ(r) is an l (l k) matri, and p is the generalized non-holonomic momentum. We can compute this momentum variable b p = l Ω T ξ where Ω T is a basis of N( ω ξ ), the null space of ω ξ. Then using () we compute the epression for p as shown in (4). ξ = A(r)ṙ + Γ(r)p T (3) p T l = Ω ξ = Ω (Iξ + IAṙ) = ` «ξ ΩI ΩIA (4) ṙ ṗ = p T σ pp(r)p + p T σ pṙ(r)ṙ + ṙ T σṙṙ(r)ṙ (5) Moreover, for sstems with a single generalized momentum variable 4, its evolution is governed b a first order differential equation 5 shown in (5), where the σ s are matrices of appropriate dimensions whose components depend solel on the base variables. Later in the paper, we will utilize both (3) and (5) and rewrite them in appropriate forms that will help us generate gaits. Eample: Now we introduce our eample sstem, the variable inertia snakeboard, which is composed of three rigid links that are connected b two actuated revolute joints as shown in Fig. (a). The outer two links have mass, m, concentrated at the distal ends and an inertia, j, while the middle link is massless. Moreover, attached to the distal ends of the outer two links is a set of passive wheels whose aes are perpendicular to the robot s links. The no sidewas slippage of these two sets of wheels provide the two non-holonomic constraints which act on the sstem. We attach a bod coordinate frame to the middle of the center link and align its first ais along that link. The location of the origin of this bod attached frame is represented b the configuration variables (, ) while its orientation is represented b the variable θ. The two actuated internal degrees of freedom are represented b the relative inter-link angles (α,α ). Hence, the variable inertia snakeboard has a five-dimensional (n = 5) configuration space Q = G M, where the associated Lie group fiber space 4 For sstems with more than one momentum variables, (5) will be a sstems of differential equations involving tensor operations as was shown in [, 5]. 5 Recall that these equations are the dnamic equations of motion along the fiber variables epressed using the generalized momentum variables.
6 Elie A. Shammas, Howie Choset, and Alfred A. Rizzi denoting the robot s position and orientation in the plane is G = SE(), the special Euclidean group. The base space denoting the internal degrees of freedom is M = S S. The Lagrangian of the variable inertia snakeboard in the absence of gravit is computed using L(q, q) = 3 i= (m iẋ T i ẋi + j i θ i ). Let L and R be the length of the middle link and the outer links, respectivel. Moreover, to simplif some epressions we assume that the mass and inertia of the two distal links are identical, that is, m i = m and j i = j = mr. Given that the fiber space has an SE() group structure, we can compute the bod velocit representation of a fiber velocit where we have ξ = T g L g ġ, that is ξ = cos(θ)ẋ sin(θ)ẏ, ξ = sin(θ)ẋ + cos(θ)ẏ, and ξ 3 = θ. sin(α R ) + sin(α ) B I = mr @ cos(α R ) cos(α ) sin(α ) + sin(α ) cos(α ) cos(α ) L +R /L sin(α ) sin(α ) «IA = mr @ cos(α ) cos(α ) A R and m = mr R R + L cos(α ) R + L cos(α ) ««sin(α) cos(α ) R + L cos(α ) R ω ξ = and ω sin(α ) cos(α ) R + L cos(α r = ) R R/ + cos(α )+cos(α ) C A(6) (7) (8) The above transformation allows us to verif the Lagrangian invariance and to compute the reduced Lagrangian. Thus, the components of the reduced mass matri as is shown in () are given in (6) and (7). Note that the reduced mass matri is not constant as was the case for the original snakeboard [3] and it depends solel on the base variables, α and α. Similarl we can write the non-holonomic constraints in bod coordinates for the variable inertia snakeboard where the components of () are given in (8). Moreover, note that the variable inertia snakeboard has a three-dimensional fiber space, SE(), and it has two non-holonomic constraints, one for each wheel set. We have verified that these constraints are invariant with respect to the group action and we know that these non-holonomic constraints are linearl independent awa from singular configurations. Thus, we conclude that the variable inertia snakeboard a mied tpe sstem. Using the above computed components of the reduced mass matri in (6) and (7) and the non-holonomic constraints in (8), we can easil compute the reconstruction equation shown in (3) and the generalized non-holonomic momentum as shown in (4). For the sake of brevit, we will not present the eplicit structure of these epressions. Finall, utilizing the generalized nonholonomic momentum and the reconstruction equations, we can rewrite the original Euler-Lagrange equations of motion in terms of the generalized nonholonomic momentum to arrive at the momentum evolution equation, (5).
3 Scaled Momentum Motion Planning for Variable Inertia Mechanical Sstems 7 Rewriting the sstem dnamics in terms of the generalized non-holonomic momentum ields a rather simple epression of the equations of motion as shown in (3) and (5). Nonetheless, we can clearl see that the generalized momentum is not zero for all time as was the case purel mechanical sstems, thus, when we integrate the reconstruction equation the second dnamic, Γ(r)p T term does not vanish. Analzing this dnamic term is rather complicated since one can not solve for the generalized momentum in a closed from. In fact, Bullo et. al. avoided this computing second term b considering gaits that nullif it. We, on the other hand, will introduce a new momentum variable that will simplif the dnamic terms analsis which will allow us to generate a richer famil of gaits. Now we manipulate (5) to a more manageable form which allows us to intuitivel evaluate the dnamic phase shift. At this point we will limit ourselves to sstems that have one less velocit constraint than the dimension of the fiber space, i.e., l k =. This assumption leaves us with onl one generalized momentum variable and forces the term σ pp (r) = in (5) as was eplained in [8]. Moreover, first order differential equation theor confirms that an integrating factor, h(r), eists for (5). Hence, the assumption l k = allows us to easil find a closed form solution for the integrating factor for the momentum evolution equation (5). In our future work, we will address the eistence of this integrating factor for sstems for which the above assumption does not hold. Thus, we define the scaled momentum as ρ = h(r)p and rewrite (3) and (5) to arrive at ξ = A(r)ṙ + Γ(r)ρ, and (9) ρ = ṙ T Σ(r)ṙ, () where Γ(r) = Γ(r)/h(r) and Σ(r) = h(r)σṙṙ (r). Now that we have written the reconstruction and momentum evolution equations in our simplified forms shown in (9) and (), we are read to generate gaits b studing and analzing the three terms, A(r), Γ(r), and Σ(r). In fact, we will use A(r) and Γ(r) to respectivel construct the height and gamma functions while we use Σ(r) to stud the sign definiteness of the scaled momentum. 4 Gait evaluation In this section, we equate the position change due to an closed base-space curve to two decoupled terms. Then, we present how to design curves in such a wa to eclusivel ensure that an of these terms is non-zero along a specified fiber direction, that is, effectivel snthesizing two tpes of gait associated with each of the decoupled terms. In the net section, we will define
8 Elie A. Shammas, Howie Choset, and Alfred A. Rizzi α F F F 3 α α α α α (a) (b) (c) G G 3 G 3 α α α α (d) (e) (f) Fig.. The three height functions, (F, F, F 3), and three gamma functions, (G, G, G 3), corresponding to the three fiber directions in bod representation, (ξ, ξ, ξ 3), for the variable inertia snakeboard are depicted in (a) through (f). The darker colors indicate the positive regions which are separated b solid lines from the lighter colored negative regions. α a partition on the space of allowable gaits such that we can generate gaits b relating position change to either one of the decoupled terms or both. For the first case, we eclusivel use the gait snthesis tools presented in this section while for the second case we define another tpe of gaits that simultaneousl utilizes both gait snthesis tools. We define a gait as a closed curve, φ, in the base space, M, of the robot. We require that our gaits be cclic and continuous curves. Having written the bod representation of a configuration velocit in a simplified manner as seen in (9), we solve for position change b integrating (9). Defining ζ as the integral of ξ and then integrating each row of (9) with respect to time we get ζ i = Z t Z Z = t ζi dt = mx Z t Φ o,j=,o<j t ξ i dt = Z t t Ā i oj(r)dr o dr j + {z } I GEO! mx Xl k A i j(r)ṙ j + Γ j(r)ρ i j dt j= Z l k X j= Γ i j(r) j= Z ṙ T Σ(r)ṙ j dt «dt {z } I DY N Note that the first term can be written as a line integral and then b using Stokes theorem we equate it to a volume integral. As for the second term we just substitute for the scaled momentum, ρ, using (). Hence, we equated position change to two integrals, I GEO which computes the geometric phase ()
Motion Planning for Variable Inertia Mechanical Sstems 9 shift and I DY N which computes the dnamic phase shift. Net, we analze how to snthesize gaits using the two independent phase shifts. 4. Evaluating geometric gaits For simplicit, we limit ourselves to two-dimensional base spaces, that is, (m = ). This allows us to equate the geometric position change contribution, I GEO, due to an gait, φ, b computing the volume integral φ F i (r,r )dr dr, where F i = Ai r Ai r s are the well-defined height function associated with the fiber velocit ξ i. Then, we generate geometric gaits b studing certain properties of the height functions: Smmetr to stud smaller portions of the base space, Signed regions to control the orientation of the designed curves as well as the magnitude of the the geometric phase shift, and Unboundedness to identif singular configurations of the robot. B inspecting the above properties of the height functions we are able to easil design curves that onl envelope a non-zero volume under a desired height function while it encloses zero volume under the rest of the height functions. For eample, Closed non-self-intersecting curves that sta in a single signed region are guaranteed to enclose a non-zero volume and Closed selfintersecting curves that span two regions with opposite signs and that change orientation as the pass from one region to another are also guaranteed to enclose a non-zero volume. On the other hand, Closed non-self-intersecting curves that are smmetric about odd points are guaranteed to have zero volume and Closed self-intersecting curves that are smmetric about even points are guaranteed to have zero volume. Note that these rules do not impose an additional constraints on the shape of the input curves. 4. Evaluating dnamic gaits Now, we will analze the second term in (), to propose gaits that ensure that I DY N is non-zero along a desired fiber direction. Note that for each fiber direction the integrand of I DY N in () is composed of the product of two terms, the gamma function, Γ i (r), and the scaled momentum variable, ρ. Thus, b analzing the Σ matri in () we propose families gaits that ensure that the scaled momentum variable is sign-definite. Then, we analze the the gamma functions in a similar wa to how we analze the height functions, that is, we stud their smmetr, signed regions, and unbounded regions. Note that, we do not use Stokes theorem on the gamma functions as we did for the height functions, since the dnamic phase shift is equated to a time definite integral not to a path integral as was the case for the geometric phase shift. Thus, b picking gaits that are located in a same signed region of Γ i (r), we ensure the integrand of I DY N is non-zero along a desired fiber direction. Eample: Now we compute the height and gamma functions for the variable inertia snakeboard. The epressions for this particular sstem are rather
Elie A. Shammas, Howie Choset, and Alfred A. Rizzi complicated and we will not present them in this paper; however, the epressions can be found in [] and we depict the graphs of the three height and gamma functions in Fig. (a) (c) and (d) (f), respectivel. These functions have the following properties which we will utilize later to generate gaits. F = G = for α = α, F 3 = G 3 = for α = α, F and G are even about both lines α = α and α = α, F and G are even about α = α and odd about α = α, F 3 and G 3 are odd about α = α and even about α = α. 5 Gait generation for mied sstem In this section, we utilize our geometric and dnamic gait snthesis to generate gaits for mied sstems. Net, we define a partition on the allowable gait space which allows us to independentl analze I GEO and I DY N and generate gaits using our snthesis tools. We respectivel label the two families of gaits as purel kinematic and purel dnamic gaits. Moreover, we propose a third tpe of gait that simultaneousl utilizes both shifts, I GEO and I DY N, to produce motions with relativel larger magnitudes. We label this famil of gaits as kino-dnamic gaits. 5. Purel kinematic gaits Purel kinematic gaits are gaits whose motions is solel due to I GEO, that is, I DY N = for all time. A solution for such a famil of gaits is to set ρ = in () which sets the integrand of I DY N to zero. Thus, we define purel kinematic gaits as gaits for which ρ = for all time. Note that for purel mechanical sstems p = ρ = b definition and for principall kinematic sstems I DY N = since p =. Hence, an gait for these two tpes of sstems is necessaril purel kinematic. However, for mied sstems, we generate purel kinematic gaits b the following two step process: Solving the scaled momentum evolution equation, (), for which ρ = ρ =. This step defines vector fields over the base space whose integral curves are candidate purel kinematic gaits. Using our geometric gait snthesis analsis on the above candidate gaits to concatenating parts of integral curves that enclose a non-zero volume under the desired height functions. Sometimes, purel kinematic gaits are referred to as geometric gaits, since the produced motion is solel due to the generated geometric phase as defined in []. Moreover, purel kinematic gaits are structurall similar to gaits proposed b Bullo in his kinematic reduction of mechanical sstems in [3].
Motion Planning for Variable Inertia Mechanical Sstems α ρ/ ma( ρ) 3 α F E D C α (a) Fig. 3. (a) Plot indicating the negative regions (lighter colored regions) of the base space where ρ <. (b) Two vector fields defined over the base space whose integral curves are purel kinematic gaits. The solid lines are integral curves of the vector fields which we will utilize to generate purel kinematic gaits. The solid dots indicate the negative regions of ρ where the vector fields are not defined. 3 G H 3 A (b) B 3 α The vector fields defined above essentiall serve the same purpose of the decoupling vector fields presented in Bullo s work. Eample: For the variable inertia snakeboard, we can easil design purel kinematic gaits b solving for the right hand side of () equal to zero. Since the right hand side of () is a quadratic in the base velocites 6, we ensure that the term ρ(α,α ) = Σ Σ Σ Σ. A plot of a ρ/ma( ρ) is shown in Fig. 3(a). The light colored regions indicate that ρ(α,α ) <, that is, we can never compute an velocities for which ρ =. In other words, we should avoid these regions of the base space while designing purel kinematic gaits. Awa from the negative regions of ρ(α,α ), we design purel kinematic gaits for the variable inertia snakeboard. The right hand side of () has four unknowns, (α,α, α, α ). Thus, at each point in the base space, that is, fiing (α,α ), we need to solve the velocities ( α, α ) for which the right hand side is zero. Since, we have two unknowns and one equation, we solve α for the ratios, α and α α for which the right hand side is zero. Thus, ignoring the magnitudes of the base velocities, the two ratios α α and α α define the slopes of vectors at each point in the base space which we use to define vector fields over the entire base space as depicted in Fig. 3(b). Hence, an part of an integral curve of the above vector fields is necessaril a purel kinematic gait. For eample, the families of lines, l = {α = α + kπ,k Z} and l = {α = α +kπ,k Z} are the simplest integral curves we could define whose velocities eactl match the above vector fields. 6 For the original snakeboard, we verified in [3] that the right hand side of the scaled momentum evolution equation is not a quadratic. This simplified the generation of purel kinematic gaits and mislead us into believing that purel kinematic gaits could be defined everwhere on the base space.
Elie A. Shammas, Howie Choset, and Alfred A. Rizzi Purel Kinematic Purel Dnamic Kino-dnamic Polgon α = π ( sin(t) ` 4 sin (t)) α = π sin(t) + cos(t) π ACEGA α = π 4 ( + sin(t) 4 sin (t)) α = π ` sin(t) + cos(t) π ` 4 Polgon α = π ( sin(3t) 5) α = π sin(t) + sin(t) π ABFECBFGA α = π 4 (sin(t) 3) ` α 6 = π sin(t) sin(t) π ` 4 Polgon α = π ( sin(t) + ) α = π sin(t) + sin(t) π 4 ACDHGEDHA α = π 3 3 ( sin(t) ) α 4 = π ` sin(t) + sin(t) π 3 3 Table. Three proposed gaits of each famil for the variable inertia snakeboard. To design a purel kinematic gait that will move the variable inertia snakeboard along sa the ξ direction, we pick an closed integral curve that will enclose a non-zero volume solel under the first height function. Using the above lines, we know that the polgon given in the first row of the first column of Table and depicted in Fig. 3(b) will move the snakeboard along the ξ direction. Similarl we construct two other polgons shown in the second and third rows of the first column of Table as depicted in Fig. 3(b) to respectivel locomote the snakeboard along the ξ and ξ 3 directions. Inspecting the above polgonal gaits, we found out that the pass through the snakeboard s singular configurations, (α,α ) = {( π, π ),( π, π )}, (Fig. (a) (c)). So rather than solving numericall for other integral curves of the vector fields and solve for other possible gaits which is a tedious process, we simpl shrunk the above proposed gaits around the center of the base space as shown in Fig. 3(b). These curves closel, but not eactl, match the vector fields. So we shall epect a change in the scaled momentum value as we traverse these gaits since the are an approimate solution. The motions of the variable inertia snakeboard due to these gaits are depicted in Fig. 4(a) (c). Note, the small magnitudes of motion due to the small volumes under the height functions. However, we can clearl see that the gaits move the variable inertia snakeboard along the and directions in Fig. 4(a) and Fig. 4(b), respectivel, and rotate the snakeboard in Fig. 4(c). Finall, recall that, our analsis is done in bod coordinates, ξ i s, which are related b the map T g L g to the fiber variables, ġ i s. Since, the fiber space for the variable inertia snakeboard is SE() which is not Abelian, the map T g L g is non-trivial; hence, one should not epect a direct correspondence between sa ξ and ẋ. This eplains the non-pure fiber motions in Fig. 4(a), where motion along ξ transforms to major motion along and minor motion along the ais. 5. Purel dnamic gaits As the name suggests, purel dnamic gaits are gaits that produce motion solel due to the dnamic phase shift, that is, I GEO = while I DY N. These gaits are relativel eas to design since these are gaits that enclose no
Motion Planning for Variable Inertia Mechanical Sstems 3 (a) 6 (b) (c) 5 4 3 3 3 4 (d) (e) (f) (g) (h) (i) Fig. 4. The actual motion that the variable inertia snakeboard will follow as the base variables follow the three purel kinematic depicted in the first column of Table shown respectivel in a, b, and c; three purel dnamic gaits depicted in the second column of Table shown respectivel in d, e, and f; and three kino-dnamic gaits depicted in the third column of Table shown respectivel in g, h, and i. The initial and final configurations for each gait are shown in gra and black colors, respectivel, while the dotted line depicts the trace of the origin of the bod-attached coordinate frame. volume in the base space. Note that all sstems that have onl one base variables have gaits that are necessaril purel dnamic, since setting m = in () will ield I GEO =. For eample, all the gaits for robo-trikke robot which was studied b Chitta et. al. in [4] are necessaril purel dnamic since there eists onl one base variable. As for sstems with more than one base space variable, it is still relativel eas to construct purel dnamic gaits. Such gaits do not enclose an area in the base space. A simple solution would be to ensure that a gait retraces the same curve in the second half ccle of the gait but in the opposite direction. Thus, we propose the following purel dnamic families of gaits: {r,r } = { n i= a i (f(t)) i,f(t)}, where f(t) = f(t + τ) is a periodic real function and
4 Elie A. Shammas, Howie Choset, and Alfred A. Rizzi a i s are real numbers. We can verif that these gaits will have zero area in the base space (r,r ). Moreover, we can verif that for the above famil of gaits, the scaled momentum variable is sign-definite, that is, ρ or ρ for all time. Then, generating purel dnamic gaits reduces to the following simple procedure: Select gaits from the above described famil and check the sign of the scaled momentum variable ρ. Analze the gamma functions depicted in (9) and () to pick the gait that ensures that the integrand of I DY N is non-zero for the desired fiber direction. Eample: For the variable inertia snakeboard, we construct three purel dnamic gaits depicted in the second column of Table. The motion due to these gaits are respectivel shown in Fig. 4(d) (f). For instance, we designed the first gait in the second column of Table such that ρ for all time. The gait is located close to the center of the base space and is smmetric about the line α = α ; moreover, onl the first gamma function is non-zero and even about the line α = α while the second and third gamma functions is odd about this line. Thus we epect a non-zero I DY N onl along the ξ direction. This motion, is largel transformed to motion along the direction as shown in Fig. 4(d). Similarl, we designed the other two gaits to move the variable inertia snakeboard along the direction, (Fig. 4(e)) and to rotate it along the θ direction, (Fig. 4(f)). 5.3 Kino-dnamic gaits Finall, we have the third tpe of gaits which we term as kino-dnamic gaits. These gaits have both I GEO and I DY N not equal to zero, that is, the motion of the sstem is due to both the geometric phase shift as well as the dnamic phase shift which are associated with I GEO and I DY N, respectivel. We design kino-dnamic gaits in a two step process. First we do the volume integration analsis on I GEO to find a set of candidate gaits that move the robot in the desired direction. The second step it to compute I DY N for the candidate gaits and verif that the effect of I DY N actuall enhances the desired motion. Essentiall, kino-dnamic gaits are variations of purel kinematic gaits. In a sense, we start b generating a purel kinematic gait but b neglecting the constraints that the gaits has to be an integral curve of the vector fields that prescribes the purel kinematic gaits. Thus, we know that scaled momentum is not necessaril zero for all time, that is, I DY N. Then, we pick the gaits for which the magnitude of I DY N additivel contribute to that of I GEO, hence, effectivel producing fiber motions with bigger magnitudes.
Motion Planning for Variable Inertia Mechanical Sstems 5 Eample: For the variable inertia snakeboard, we can generate kinodnamic gaits b using the volume integration analsis to produce candidate gaits. For eample, to generate a gait that rotates the variable inertia snakeboard in place, we start b designing a curve in the base space that envelopes a non-zero volume onl under the third height function of the variable inertia snakeboard (Fig. (c)). A figure-eight tpe curve with each of its loops having opposite orientation and ling on the opposite side of the line α = α, will envelope non-zero volume onl under the third height function. This curve is the last curve in the third column of Table. We simulated this proposed gaits and indeed it does rotate the variable inertia snakeboard along the θ direction as shown in (Fig. 4(i)). Similarl, we designed two other curves depicted in the first and second rows of the last column of Table to move the variable inertia snakeboard along the direction, (Fig. 4(g)), and the direction,(fig. 4(h)). In this section we have generated three of each tpe of gaits that moved the variable inertia snakeboard in an specified global direction. Moreover, we have the freedom to choose from several of the tpes of gaits that we have proposed earlier. It is worth noting that the purel kinematic and purel dnamic gaits were the easiest to design since we are eclusivel analzing either I KIN or I DY N and not both at the same time as is the case for kindnamic gaits. 6 Conclusion In this paper, we studied mied non-holonomic sstems and designed three families of gaits, purel kinematic, purel dnamic, and kino-dnamic gait, to move such sstems along specified fiber directions. This work is a generalization over our prior work where we used one tpe of the gaits defined here to analze two other sstems, purel mechanical and principall kinematic. Moreover, our technique affords better fleibilit in choosing the parameters of the suggested gaits and reduces the need for intuition about how to to manuall adjust these parameters. One of the contributions of this paper is the introduction of the scaled momentum variable which greatl simplified our gait generation analsis. This new variable allowed us to rewrite both the reconstruction as well as the momentum evolution equation in simpler forms that are suitable for our gait generation techniques. Another contribution is the introduction of the novel mechanical sstem, the variable inertia snakeboard. This sstem is similar enough to the original snakeboard that we can relate our results to this well known sstem, but at the same time it did not over simplif the gait generation problem. In fact, through analzing the variable inertia snakeboard, we identified regions in the base space where purel kinematic gaits are not possible. There are no such regions for the original snakeboard. This paper constitutes a first step towards developing an algorithmic gait snthesis technique. Our net step would be to rela the assumptions we made
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