sagittal plane only. Each stride is governed by two sets of equations of motion which can be derived from the Lagrangian equations; one for stance phase (time ...
Dynamic Stability of Variable Stiffness Running Jae Yun Jun and Jonathan E. Clark Department of Mechanical Engineering FAMU/FSU College of Engineering Tallahassee, FL 32310, USA {jaeyun,clarkj}@eng.fsu.edu
Abstract— Humans and animals adapt their leg impedance during running for both internal(e.g. loading) and external(e.g. surface) changes. In this paper we examine the relationship between leg stiffness and the speed and stability of dynamic legged locomotion. We utilize a torque-driven reduced-order model of running based on a successful family of running robots to show how optimal clock-driven controllers can interact with variably compliant limbs to adapt to changing operating conditions. We show that the leg stiffness adaptation gives, in general, better results than simply optimizing the gait controller and nearly as good as the co-optimization of controller and leg stiffness.
I. INTRODUCTION Running animals utilize their legs to run effectively over a large range of terrains. With each step the gravitational and kinetic energy of their body is transformed via their leg muscles, tendons and ligaments into strain energy which is stored during the deceleration in the first half of stance and is returned, aided by muscle contraction, during the second half of stance to re-accelerate the body[1]. These dynamics of running can be modeled using the Spring Loaded Inverted Pendulum (SLIP) model[2],[3] and despite its simplicity it accurately captures the ground reaction forces and the motion of the center of mass for a wide scope of animals. This model has, in turn, led to the development of a range of dynamically running robots including Raibert’s hoppers[4], Scout[5], Sprawlita[6], and RHex[7]. Despite the success of these robots, their performance, especially in variable terrains, pales in comparison to their biological inspirations. The compliant and viscoelastic legs of these animals allow them to run over a large variety of terrains while adjusting their leg stiffness[8]. Apparently these animals can operate at or near optimal conditions for passive, dynamic self-stabilization because of the viscoelastic properties of their passive mechanisms, known as preflexes, stabilize their locomotion from perturbations[9]. Generally, we are interested in designing legs which can mimic or exceed the performance of animal legs by understanding the animal legs’ functional properties and by relating their passive properties to the controller design and ultimately to the robot’s dynamic performance. Several robotic designers have attempted to imbue their devices with mechanically adaptable impedance properties to provide both energetic efficiency and the flexibility to deal with changing conditions. Variable stiffness limbs have been developed for walking[10],[11] and more recently for a bipedal
Fig. 1: RespondBot RDK[15], a hexapod robot considered for studying the effect of the legs with variable stiffness on the robot’s dynamic stability
runner[12],[13]. Although biological precedent and basic dynamic systems theory suggest that tuned resonant running should improve the performance of these systems[14],[12], no robot has yet demonstrated this advantage in terms of speed or stability for running. Recently Galloway et alt.[16] have developed a lightweight variable stiffness leg for a small hexapedal running robot called RespondBot RDK(see Fig. 1). RespondBot, like its predecessor, RHex[7], utilizes six compliant legs and a tasklevel open loop controller to run at several body lengths per second and over rough terrain. The controller for this robot, referred to as a ’Buehler-clock’ prescribes a reference pattern for the leg during stance and re-circulation[17]. As this six parameter controller is difficult to tune, the best gaits for RHex have been derived through costly on-board optimization trials[18]. In ongoing research we are investigating the effect of cooptimizing the controller and leg stiffness on the performance of this, the newest member of the RHex family. Although our efforts here are motivated by, and anchored to, this platform, we are fundamentally interested in the larger question of how the interplay between the controller and the passive leg properties affects a system’s dynamic performance and stability. In order to pursue this in a more general environment we utilize reduced-order dynamic models. In addition to inspiring
W hip
]
m k
W hip ]o
\
b \
Fig. 2: Dynamical Model, a modified version of SLIP model with a torque actuator added at the hip. The leg angle, ψ, is defined to be positive counterclockwisely.
robotic designs, these SLIP-like models continue to yield insight into running such as, how the effectively non-linear spring properties of articulated (knee-based) leg designs can aid in stabilization[19], and how appropriate feed-forward injection of energy into the stride can increase a gait’s robustness to disturbances[20]. In this paper we introduce a torque-driven SLIP-like hopper with a controller modeled strictly after the one implemented on RespondBot RDK, and show that significant increases in the optimal performance of the runner can be achieved under changing operating conditions by utilizing variable stiffness limbs. Section II introduces the dynamical model, the RHex-based controller design, the optimization techniques used to find the most stable gaits and the stability analysis used in the objective function. Section III describes the relations between dynamic stability, leg stiffness and alterations to the robot’s physical parameters, and Section IV summarizes the conclusions and gives a brief discussion of future work. II. M ETHODS A. Dynamical Model The running model utilized in this work is shown in Fig. 2 and is a variant of the standard SLIP model which has been adapted to emulate the actuation scheme utilized in the RespondBot robot. As with the SLIP model, the robot body is represented by a point mass and the leg as a massless translational spring with constant stiffness. To account for the energetic losses in the leg a linear viscous damper is added in parallel to the spring. The foot contact is modeled as a pin joint and no foot slip is allowed. The unique actuator in the system, to compensate the energy loss, is rotational and is located on the hip. The motion of the robot is considered to be constrained to the sagittal plane only. Each stride is governed by two sets of equations of motion which can be derived from the Lagrangian equations; one for stance phase (time during which the foot remains in contact with the ground) and the other, for flight phase (time during which the foot is not in contact with the ground). The first set
of equations is k (ζ − ζo ) − ζ¨ = ζ ψ˙ 2 − g cos ψ − m ˙ ψ˙ τ g 2 ζ + sin ψ + hip2 ψ¨ = ζ
ζ
b ˙ mζ
(1)
mζ
where ζ, ψ, g, k, m, b, and τhip denote leg length, angle of the leg with respect to vertical, gravitational acceleration, leg stiffness coefficient, robot body mass, leg damping coefficient and the torque applied on the hip, respectively. The second set of equations is x ¨=0 z¨ = −g (2) ψ¨ = 0 where x, z, ψ and g represent the position of the body in the horizontal axis, position of the body in the vertical axis, angle of the leg with respect to vertical and gravitational acceleration, respectively. In this model, the leg is massless and, therefore, no leg inertia is present. Under this assumption, the leg perfectly follows the trajectory dictated by the controller when it recirculates during flight phase. In order to complete the hybrid dynamics we define two transition events: leg touch-down and lift-off. The touch-down event occurs when the leg touches the ground, i.e. when the condition z = ζ cos ψ is satisfied. The lift-off event occurs when the vertical ground reaction force becomes zero. TABLE I: PHYSICAL PARAMETERS FOR DYNAMICAL MODEL Parameters Symbols Values Units Body mass m 2.5 kg N/m Leg stiffness k 2, 000 a 0.13 m Leg rest length ζo Leg damping constant b 24.77 a N s/m 0.0436 Nm Motor Stall torque τs 834.616 rad/s Motor No load speed ωnls 24 Gear ratio gr Gravity acceleration g 9.81 m/s2 a
effective value for three legs in parallel for a tripod
In addition, we considered a simple motor model taking into account the specifications of the actual motor and gearbox utilized on each RespondBot hip. With this model, the leg angular speed and hip torque are restricted by the idealized torque-speed curve. Finally, the physical parameters utilized in the simulation model correspond to the actual RespondBot physical values as shown in Tab. I. We used the RespondBot C-type leg torsional stiffness coefficient explained in [16] and the torsional damping coefficient is computed from the decaying ratio of the leg vibration captured using a high-speed camera and taking into account the leg torsional stiffness and its mass. The conversion from the torsional values to translational values is done in a
S
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\B tB
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0
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T tB
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Fig. 3: Monopod controller. It consists of a PD controller with desired leg trajectory characterized by two slopes. The flatter slope corresponds to slow stance phase and the steeper, the fast swing phase.
Fig. 4: Biped controller. The desired trajectories of two legs are represented by unbroken line segments for the first leg’s desired trajectory and by line segments of long and short dashes for the second leg.
similar manner as in [19]. In this paper, we are interested in the variations of the leg stiffness only and therefore we kept the non-dimensional damping ratio fixed at 0.16.
assumed symmetry for both legs. The difference is that the driving period is now defined as
B. Controller We used the same controller as RespondBot does locally on each of six hips, a PD controller where the desired leg angular position and speed are dictated by a periodic function known as Buehler clock[17](see Fig. 3). In this paper, we chose a different, but equivalent, set of parameters from the ones that Saranli et alt.[17] defined. These parameters are ψA , ψB , tB and T representing the desired touch down leg angle, desired lift off leg angle, desired lift off time and the desired stride period, respectively. As we can see from Fig.3, a single stride driven by the controller is composed by a slow and a fast phase, roughly corresponding to the stance and flight phases of the robot[17]. The leg during the flight, or recirculation, phase must progress above the hip (unlike biological systems) due to the reduced kinematics of the leg design. The slow phase starts at ψ = ψA which lasts for tB seconds. Then, the fast phase starts at ψ = ψB which lasts until the end of the period, at T seconds. Finally, the description of the PD controller is completed with two additional parameters: the proportional gain, kp , and the derivative gain, kd . From the above description, the control input signal, the commanded hip torque (τhip ), can be generated as, τhip
˙ = kp (ψdes − ψ) + kd (ψ˙ des − ψ)
(3)
where ψ and ψ˙ are the actual leg angular position and angular speed, respectively. ψdes and ψ˙ des are defined as ψB −ψA tlocal + ψA , 0 ≤ tlocal < tB tB ψdes = a · t tB ≤ tlocal < T ψB −ψA local + b, (4) , 0 ≤ t < tB local tB ψ˙ des = a, tB ≤ tlocal < T B where a = ψA −2π−ψ , b = ψB −a·tB and tlocal is the congruent T −tB modulo T of the time, t. All angles are defined to be positive counterclockwisely. The same parameters utilized to describe the monopod controller can be used for the biped controller because we
T = 2tB + 2tFF
(5)
where tB , tFF and T denote the duration of the slow stance phase, duration of the double fast swing phase, and the driving period, respectively (see Fig. 4). The first leg starts to move from the beginning of slow stance phase with ψ = ψA while the second leg is in fast swing phase with its leg position and velocity corresponding to the commanded trajectories at t = 0 (see Fig. 4). When the first leg abandons the stance phase (and therefore enters the flight phase), the second leg must be at ψ = ψC . The duration of double fast swing phase, tFF , is given by the duty factor, the fraction of time corresponding to the slow stance phase. The occurrence of double support is avoided by imposing the condition tLO local ≤
T 2
(6)
where tLO local is the congruent modulo T of the lift-off time, tLO . As we are only interested in modeling running gaits, if the condition (6) is not satisfied, then the corresponding gait is discarded. C. Periodic gait and dynamic stability analysis As is commonly done, we studied the stability of periodic gaits by looking at the eigenvalues of the Jacobian matrix of the Poincare map about its fixed points, see for example [20],[21]. Fixed points in the Poincare map correspond to periodic orbits in the flow in the phase space. We defined our Poincare map by taking a Poincare section to the flow at the instant of touch down. At this instant, the six-dimensional state of the system is reduced to three dimensions. We chose (vTD , δTD , ψTD ) to be the states to describe the system at touchdown, where vTD , δTD , and ψTD are the magnitude of the velocity of the center of mass (CoM) at touch-down, angle of the velocity of the CoM at touch-down and leg angle with respect to vertical at touch-down, respectively. If any of the eigenvalues is larger than unity, then the gait is unstable. If
any of the eigenvalues is unity, then the corresponding gait is neutrally stable. Finally, if all of them are less than unity, then we have an asymptotically stable gait. The fixed points for each permutation of controller and physical parameters were found via a Newton-Raphson based iterative method. For this system for each unique set of physical and controller parameters, there is at most one fixed point, i.e., one periodic gait. D. Optimizer For a given set of physical parameters it is possible to find a wide range of stable periodic gaits by varying the controller parameters. During our simulations we observed a discrete family of gaits among which there existed a strong correlation between the eigenvalues of the Jacobian matrix of their Poincare maps and their forward body velocities. Our goal is to search for the controller parameters that optimize the dynamic stability (that is, minimize the largest eigenvalue of the linearized Poincare Map) taking into account the average forward body velocity within a stride in order to find fast, SLIP-like motions and avoid slow or shallow gaits. On the other hand, we penalized the unstable gaits (eigenvalues larger than unity), so as to guarantee to have fast stable gaits. Considering this, our optimization problem is defined as min f (p)
space for which there exists a periodic gait. For this we used brute force method, forming a grid of uniformly distributed points in the feasible region in the state space and found an initial state, (vTD , δTD , ψTD ), that converges to the fixed point. If no periodic gait is found, then a default large value was assigned to the corresponding cost. The second difficulty was to determine the size of the initial simplex to ensure convergence. Acceptable initial simplexes were identified via an iterative heuristic procedure. TABLE II: STATES Symbols m k ψA ψB T tB kp kd voTD δoTD ψoTD
p
a
s.t.
b c
˙ x(t) = g(x(t), u(t), p) x(t+LO ) = m(x(t-LO )) x(t+TD ) = n(x(t-TD )) x(tTD ) = xo r(x(t), u(t), p) ≥ 0 s(x(t), u(t), p) = 0
where x ∈ 6 is the state vector, p ∈ 6 is the controller parameters and u ∈ 3 is the input where the unique non-zero component corresponds to the leg angle axis. The objective function is f (p) = |λmax (J(p)) |/v f orward , with J denoting the Jacobian matrix of the Poincare map([20],[21]), and v f orward being the average forward body velocity within a stride. g are the equations of motion, (m, n) are the boundary conditions, xo are the initial conditions, and (r, s) are the constraints on the variables. As in previous research in solving this type of optimization problem[22],[23], we also chose to utilize a Nelder Mead algorithm, a direct search method, to solve this problem because, as Mombaur et alt. noted, the objective function is non-differentiable and possibly even non-Lipschitz if the Jacobian matrix of the Poincare Map is not symmetric[22]. We implemented the modified version of Nelder-Mead algorithm explained in [22], except that for the termination condition, we used the norm-2 of the vector differences between the cost associated to each vertex that form the simplex in question and the cost corresponding to the centroid of the simplex. There were two aspects that made using this algorithm difficult. The first difficulty was to find, for a given pair leg stiffness and body mass, an initial vertex in the controller
RANGE FOR PARAMETERS AND INITIAL Meaning body mass leg stiffness leg TD anglec leg LO anglec stride periodc stance durationc proportional gain derivative gain body vel. mag. body vel. ang. leg TD angle
Range [2, 4] a [1200, 3600]b [0, 90] [−90, 0] [0.1, 0.6] [0.1T, 0.6T ] [0.01, 100] [0.01, 100] [0.5, 2.5] [−30, −1] [10, 70]
Units kg N/m deg deg s s N m/rad N m s/rad m/s deg deg
0.5kg of separation between two consecutive body mass values 400N/m of separation between two consecutive leg stiffness desired leg trajectory parameters
III. RESULTS AND DISCUSSION We performed simulations of the modified torque driven SLIP model with a ’Buehler-clock’ controller in Matlab. The equations of motion are solved numerically using ode23s with precisions altol = 10-9 and reltol = 10-6 . Table II specifies the ranges of body mass and leg stiffness values considered, the feasible region in the controller space for the optimization problem and the ranges of initial states (voTD , δoTD , ψoTD ) considered for the brute force method implemented for identifying fixed points. To examine the effect of changing operating conditions on the running performance we considered the case of the robot changing mass. In practice this would occur when the robot either picks up or deploys some sort of payload. In this section we considered cases where mass of the nominally 2.5kg robot varied between 2.0 and 4.0 kg, as listed in Table II. A. Optimizing controller over body-mass & leg-stiffness plane First, for each pair of body mass and leg stiffness within the specified ranges, we ran a full Nelder-Mead optimization in order to identify the optimal controller parameters. The resulting cost surface and contour are shown in Fig. 5(a) and Fig.5(c). We note that the more we move toward the left and top corners of Fig. 5(a) and Fig. 5(c), the cost values increase because their corresponding eigenvalues get larger; that is,
(a)
(b)
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0.55
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0.45
body mass(kg))
body mass(kg))
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2 1200
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(d)
Fig. 5: Stability analysis: (a) cost-surface obtained by optimizing the controller for each pair of body mass and leg stiffness; (b) cost-surface obtained by using the controller optimized only for the nominal physical parameters over the ranges of body mass and leg stiffness considered; (c) and (d) cost-contour corresponding to (a) and (b) where the gray scale represents eigenvalue over average forward body velocity
the gaits become less stable. In general, the cost surface declines for larger values of stiffness and lower mass because the corresponding average forward velocity is higher. This is because the corresponding natural frequencies get larger. However, at the right and bottom corner of the cost surface and contour, the cost increases rapidly because the system’s natural frequency is too large and the optimizer cannot find any fast, SLIP-like gaits. From Fig. 5(a,c) we note that as body mass increases, in order to keep the cost small, we also need to increase the leg stiffness.
For configurations removed from the center of the valley, the periodic gaits become less stable, until we find cases of unstable periodic gaits. For the cases where the gait is unstable we display a cost = 0.8 in order to help visualize the details of the surface where the gait is stable. In general, we observe that the stability results obtained using a fixed controller, optimized for one particular physical setting, are close to those obtained by optimizing the gait over body mass and leg stiffness plane. This suggests the relative insensitivity of the optimal control parameters to changes in mass or leg stiffness.
B. Fixed controller over body-mass and leg-stiffness plane
C. Different strategies to optimize stability
In this section, we analyze performance using the controller optimized for the physical values of leg stiffness and body mass values, k = 3,200N/m and m = 2.5kg. The corresponding cost surface and contour are shown in Fig. 5(b) and Fig. 5(d), respectively. As before, here we also have a valley with a clear minimum cost orientation in the stiffness-mass plane.
Finally, as shown in Fig. 6, we examine different strategies for maximizing performance as we vary the body mass. For the first, and nominal, case we used the fixed controller optimized for the nominal physical parameters and fixed leg stiffness. These simulations suggest that even without any changes to the system the robot should be able to run stably for most of body
and compare the results with the experimental data in order to validate the models.
0.39 0.38
|Omax |/ vforwward
R EFERENCES 0.37 0.36 0.35 0.34 0.33
no adaptation controller adaptation stiffness adaptation controller+stiffness adaptation
0.32 0.31 2
2.5
3
3.5
4
b d mass(kg) body (k )
Fig. 6: Stability analysis for different optimization strategies. For m = 2kg, the data point for ’no adaptation’ is removed because the gait is unstable (giving a very high cost value) and for ’controller adaptation’ its cost value is high because it corresponds to a remarkably slow gait. mass values considered except when m = 2kg for the reason explained in the previous section. The second case shows improved stability results when the controller, but not the leg stiffness, can be altered. This corresponds to the standard RespondBot design – that is without variable stiffness legs. For m = 2kg the corresponding cost is high because the gait is less stable. The third case is when we only varied the leg stiffness values leaving the controller fixed to the optimal settings found for the nominal physical parameter values. Finally, in the fourth case, the optimization is done over controller space and a range of leg stiffness values are considered. As expected, the co-optimization of controller and stiffness gives the most optimal performance results for all body mass values considered while the no adaptation case gives the poorest results. But surprisingly, the leg stiffness adaptation gives better results than that of controller adaptation for the values of body mass considered. IV. CONCLUSION In this paper, we have shown that as the robot’s body mass varies varying the leg stiffness alone, without changing the controller adapted for a particular set of physical parameters, gives stability results in general better than those obtained by optimizing the controller alone and nearly as good as those obtained by co-optimizing the controller and the leg stiffness. This result matches the hypothesis of [14],[12], that is, tuned resonant running improves the performance of the robots. Also, this fact affirms our belief that proper mechanical design exploiting mechanical preflexes is critical before starting to design and optimize the controller. In the near future we will attempt to validate these findings via experimental tests on the physical robot. In addition, we would like to consider the effect of other control schemes and dynamical models for legs such as an articulated leg model with non-linear spring properties
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