Figure - uploaded by Sean Carver
Content may be subject to copyright.
State space and trajectories of the SLIP with motion constrained to a vertical line. The state variables under this constraint are vertical position of the body (vertical axis) and vertical velocity of the body (horizontal axis). The four quadrants represent the four phases of motion of the SLIP (in order from apex: descent, compression, decompression, and ascent). The hopper is in flight when the vertical position of the body is greater than one (the uncompressed leg length). The quadrant boundaries are the event conditions (touchdown, bottom, lift-off, and apex, respectively). The thick vertical line (in flight, vertical velocity 0) is the Poincaré section defining apex, parametrized by a single variable (vertical body position at apex or hopping height). We assume that trajectories start and stop on this section. In this example, the controller acts by changing the decompression spring constant at the bottom. Trajectories with different possible controls diverge from this point. Some reach subsequent apex states that are too low (red), some that are two high (blue), and for one particular value of the control (the deadbeat control), the subsequent apex state reaches its desired height (green). If the initial height were different (but within some range), the deadbeat control would also be different, but would still exist (not shown).
Source publication
This paper investigates the control of running gaits in the context of a spring loaded inverted pendulum model in three dimensions. Specifically, it determines the minimal number of steps required for an animal to recover from a perturbation to a specified gait. The model has four control inputs per step: two touchdown angles (azimuth and elevation...
Contexts in source publication
Context 1
... class of models to which our methods apply consists of those that can be written as an iteration of a discrete-time return or Poincaré map r 1 ; see Fig. 2. This map describes locomotion as the evolution of a sequence of state-vectors x j . These vectors are indexed by the step-number j. In our case, x j is the state of the SLIP at the jth apex; x j X. In a more general context, the components of each state vector in the sequence might be positions and velocities of each body seg- ment ...Context 2
... x j to x j . When New- ton's method fails to converge, the perturbed state x j is re- placed with the midpoint of the segment joining x j to x j . The replacement is closer to reference than the initial perturbed state making Newton's method more likely to converge. If The vertical axis is the applied control decompression spring constant, see Fig. 2. The blue dashed curves represent the level sets of the Poincaré return map the locus of initial apex states and controls which send the hopper to a point with a particular apex height on the next step. The thick blue curve is the level set corresponding to the desired final height. In other words, if the control or decompression ...Context 3
... to a point with a particular apex height on the next step. The thick blue curve is the level set corresponding to the desired final height. In other words, if the control or decompression spring constant is chosen so that the corresponding point on this graph lies on the thick blue curve, then the subsequent apex has the desired height, see Fig. 2. Note that only the vertical location on the graph can be chosen by the controller-the horizontal location is determined by the initial apex height. Thus the thick blue curve is the graph of the deadbeat control law deadbeat control as a function of the initial apex state. The asterisk is the reference point that defines the ...Citations
... This insight was generalized by Seyfarth [45], initiating a body of "swing-leg retraction" literature (e.g., [46,47]) that brought about two fundamental observations that bear on our work. First, minimally sensed stabilization is not only achievable by control on hybrid transitions [48,49], but can afford deadbeat performance as well with only a bit more sensing (here, deadbeat control refers to a strategy resulting in exact correction to perturbations in a finite -typically minimum -number of steps [50]). Specifically, as shown numerically [51] and analytically [52], proper feedforward servoing of sagittal leg angle in flight affords control over the apex height with no sensing required other than the detection of the apex and touchdown events, even when running over uneven terrain. ...
... Specifically, as shown numerically [51] and analytically [52], proper feedforward servoing of sagittal leg angle in flight affords control over the apex height with no sensing required other than the detection of the apex and touchdown events, even when running over uneven terrain. Second, the implicit function theorem provides sufficient conditions for the existence of deadbeat control given a sufficiently expressive input vector using full state feedback [50]. Studies in humans [53] and birds [54][55][56] document some combination of feedforward and feedback hybrid transition control strategies during biological running, further motivating their study by roboticists. ...
This paper develops a three-degree-of-freedom sagittal-plane hybrid dynamical systems model of a Groucho-style bounding quadrupedal run. Simple within-stance controls using a modular architecture yield a closed-form expression for a family of hybrid limit cycles that represent bounding behavior over a range of user-selected fore-aft speeds as a function of the model’s kinematic and dynamical parameters. Controls acting on the hybrid transitions are structured so as to achieve a cascade composition of in-place bounding driving the fore-aft degree of freedom, thereby decoupling the linearized dynamics of an approximation to the stride map. Careful selection of the feedback channels used to implement these controls affords infinitesimal deadbeat stability, which is relatively robust against parameter mismatch. Experiments with a physical quadruped reasonably closely match the bounding behavior predicted by the hybrid limit cycle and its stable linearized approximation.
... A prototypical trajectory for the SLIP hopper is shown in Figure 2. For the full dynamical equations, we refer the reader to [33], whose formulation we used for this work. ...
One of the fundamental challenges in realizing the potential of legged robots is generating plans to traverse challenging terrains. Control actions must be carefully selected so the robot will not crash or slip. The high dimensionality of the joint space makes directly planning low-level actions from onboard perception difficult, and control stacks that do not consider the low-level mechanisms of the robot in planning are ill-suited to handle fine-grained obstacles. One method for dealing with this is selecting footstep locations based on terrain characteristics. However, incorporating robot dynamics into footstep planning requires significant computation, much more than in the quasi-static case. In this work, we present an LSTM-based planning framework that learns probability distributions over likely footstep locations using both terrain lookahead and the robot's dynamics, and leverages the LSTM's sequential nature to find footsteps in linear time. Our framework can also be used as a module to speed up sampling-based planners. We validate our approach on a simulated one-legged hopper over a variety of uneven terrains.
... For systems that demonstrate nonlinear step-to-step dynamics, one can use numerical root finding to find a deadbeat controller. For example, the step-to-step dynamics of the 3 dimensional spring-loaded inverted pendulum is non-linear and one can find foot placement angle and spring stiffness that enables two-step deadbeat control [26]. For more complex systems, simple models help compute control inputs to enable deadbeat control, and then map them to joint torques using inverse kinematics and/or inverse dynamics [27]. ...
For bipedal robots to walk over complex and constrained environments (e.g., narrow walkways, stepping stones), they have to meet precise control objectives of speed and foot placement at every single step. This control that achieves the objectives precisely at every step is known as one-step deadbeat control. The high dimensionality of bipedal systems and the under-actuation (number of joint exceeds the actuators) presents a formidable computational challenge to achieve real-time control. In this paper, we present a computationally efficient method for one-step deadbeat control and demonstrate it on a 5-link planar bipedal model with 1 degree of under-actuation. Our method uses computed torque control using the 4 actuated degrees of freedom to decouple and reduce the dimensionality of the stance phase dynamics to a single degree of freedom. This simplification ensures that the step-to-step dynamics are a single equation. Then using Monte Carlo sampling, we generate data for approximating the step-to-step dynamics followed by curve fitting using a control affine model and a Gaussian process error model. We use the control affine model to compute control inputs using feedback linearization and fine tune these using iterative learning control using the Gaussian process error enabling one-step deadbeat control. We demonstrate the approach in simulation in scenarios involving stabilization against perturbations, following a changing velocity reference, and precise foot placement. We conclude that computed torque control-based model reduction and sampling-based approximation of the step-to-step dynamics provides a computationally efficient approach for real-time one-step deadbeat control of complex bipedal systems.
... We assume that the output of MPC is perfectly tracked by next controllers. [Zaytsev 2015, Kajita 2003, Carver 2009]. Following the concept of Capturability ], we can guarantee that the robot is able to stop in a few steps without falling (for whatever horizon length -covering whatever number of steps) and this is enough to guarantee that it is able to simply avoid falling. ...
This work focuses on two challenging tasks for humanoid robots: bipedal balance and collision avoidance in a dense crowd. We solve these tasks on a limited time horizon in which we can anticipate the consequences of robot actions.
We can guarantee that the robot is able to stop in a few steps and avoid falling. When the robot is not planning to stop but to continue walking, we show the guarantee to avoid falling is not lost but it depends on the length of the time horizon. It is impossible to know beforehand what people will do next, so we cannot guarantee that no collision will ever occur. Over a limited time horizon we can guarantee Passive Safety: the robot is able to stop before a collision occurs. This safety guarantee is combined with fall avoidance in a Model Predictive Control scheme. The capacity for the robot to react and avoid collisions is constrained once a step is planted on the ground, until the next step is initiated. With the control scheme outlined above the robot reacts not only at each step initiation but also in between. We show that reacting only once per step (thus saving computational power) does not degrade collision avoidance capability.
The time left for people to react and avoid collisions once the robot has stopped (to guarantee Passive Safety) might not be enough. We propose a new control scheme called Collision Mitigation that guarantees fall avoidance while aiming to leave as much time as possible for the people to react. As a result, the robot collides less often and later than when it guarantees Passive Safety. This scheme can be adapted to take different priorities into account. For example, when the members of a crowd are divided in robots and people, the robot should leave as much time as possible for the people to react and then, if possible, for the other robots. Or when the robot must reach a target location at the utmost important and people might obstruct the motion of the robot, the robot can jostle people if necessary to reach the location.
... Specifically, as shown numerically [222] and analytically [59], proper feed-forward servoing of sagittal leg angle in flight affords control over the apex height with no sensing required other than the detection of the apex and touchdown events, even when running over uneven terrain. Second, the implicit function theorem provides sufficient conditions for the existence of deadbeat control given a sufficiently expressive input vector using full state feedback [48]. Studies in humans [38] and birds [35,63,64] Noting that previous work controlling hybrid transitions in legged locomotion has been limited to varying the flight leg angle, we take inspiration from Blickhan's studies indicating that humans vary both their leg angle and leg length in flight to affect touchdown conditions [99,160] and expand consideration of hybrid transition control to vary flight leg angle and length. ...
... Proposition 2. The maps H and S have a fixed point at: 48) and ...
... Here, deadbeat control refers to a strategy resulting in exact correction to perturbations in a finite (typically minimum) number of steps[48]. ...
How does a robot's body affect what it can do? This thesis explores the question with respect to a body morphology common to biology but rare in contemporary robotics: the presence of a bendable back. In this document, we introduce the Canid and Inu quadrupedal robots designed to test hypotheses related to the presence of a robotic sagittal-plane bending back (which we refer to as a "spine morphology"). The thesis then describes and quantifies several advantages afforded by this morphological design choice that can be evaluated against its added weight and complexity, and proposes control strategies to both deal with the increase in degrees-of-freedom from the spine morphology and to leverage an increase in agility to reactively navigate irregular terrain. Specifically, we show using the metric of "specific agility" that a spine can provides a reservoir of elastic energy storage that can be rapidly converted to kinetic energy, that a spine can augment the effective workspace of the legs without diminishing their force generation capability, and that -- in cases of direct-drive or nearly direct-drive leg actuation -- the spine motors can contribute more work in stance than the same actuator weight used in the legs, but can do so without diminishing the platform's proprioceptive capabilities. To put to use the agility provided by a suitably designed robotic platform, we introduce a formalism to approximate a set of transitional navigational tasks over irregular terrain such as leaping over a gap that lend itself to doubly reactive control synthesis. We also directly address the increased complexity introduced by the spine joint with a modular compositional control framework with nice stability properties that begins to offer insight into the role of spines for steady-state running. A central theme to both the reactive navigation and the modular control frameworks is that analytical tractability is achieved by approximating the modes driving the environmental interactions with constant-acceleration dynamics.
... The equations of motion are non-integrable through the entirety of the gait cycle, so a closed form analytic solution to fully describe the motion does not exist. Optimization and simulation techniques are commonly used to generate numeric solutions and to find stable parameter combinations that simulate SLIP running (Birn-Jeffery et al., 2014;Blickhan, 1989;Carver, Cowan, & Guckenheimer, 2009;Ludwig et al., 2012;Seyfarth et al., 2002). To more easily apply the model, approximations have been made to simplify the computation, including modeling the vertical GRF as a sinusoid (Blum, Lipfert, & Seyfarth, 2009;Morin et al., 2005;Robilliard & Wilson, 2005), assuming a small angle-sweep (Geyer, Seyfarth, & Blickhan, 2005), and isolating analysis at midstance (Farley et al., 1993;Ferris, Louie, & Farley, 1998). ...
Running is fundamentally a simple activity, but the physical realization of it is complex. The gait patterns of a runner are the product of ever-changing systems and interactions of biomechanical components, and as such, the study of these mechanical characteristics is challenging. Traditional methods have focused on discrete components of gait and thus struggle to contextualize observations. Systemic analyses have been limited to simple descriptive models, often with exclusive or restrictive assumptions. This dissertation sought to develop novel methods for the systemic analyses using an established canonical model of the running gait – the spring-mass model – as a template. It further sought to conduct a series of biomechanical studies using this template-based approach as a framework to interpret the observations. Specifically, a method is first presented to estimate the system-level spring-mass characteristics of a runner using nonlinear regression with only the vertical ground reaction force time series of the runner. To facilitate this method, a novel parameterized form of the sinusoidal vGRF approximation was derived and validated. This NLR-based analyses yielded leg stiffness estimates that were consistent with traditional methods and further suggested that additional systemic parameters do not behave as traditional methods assume. Next, two investigations are presented that explore this method along with new methods for spring-mass dynamics comparisons and with established methods for spring-mass parameter analysis. These investigations included a cohort comparison of elite Kenyan distance runners against a cohort of non-elite recreational runners and a paired comparison of subjects before and after an ultramarathon. It was shown that the Kenyan runners behaved more like the simple elastic system than the recreational runners and that the ultra-marathon runners demonstrated consistent systemic patterns but greater overall template dissimilarity following the race. Finally, traditional methods of spring-mass analyses were applied with a more comprehensive mixed-model experimental design to fully characterize the system-level behavior of elite middle distance runners across a spectrum of speeds. The mixed-model template-based analysis revealed that the elite runners ran as stiffer systems than their sub-elite counterparts and that their mechanical behavior was more persistent across speeds. Together, this series of investigations established and validated new methods and improved upon the implementation of existing methods with which to assess running gait holistically and analyze it as a system. It is hoped that this work will provide useful tools, new frameworks, and fresh inspiration for scientists, coaches, and athletes to assess and interpret the movements of runners.
... Specifically, as shown numerically [35] and analytically [36], proper feed-forward servoing of sagittal leg angle in flight affords control over the apex height with no sensing required other than the detection of the apex and touchdown events, even when running over uneven terrain. Second, the implicit function theorem provides sufficient conditions for the existence of deadbeat control given a sufficiently expressive input vector using full state feedback [34]. Studies in humans [37] and birds [38]- [40] document some combination of feed-forward and feedback hybrid transition control strategies during biological running, further motivating their study for roboticists. ...
... Section II-C introduces dynamical simplifications in the form of Approximations 2, 1, and (33) that -together with the previous modeling choices -give the cascaded system the trivial dynamics depicted in Figure 4. These modeling and control choices yield simple closed form expressions for the flow on the hybrid modes (34), (35), that in turn allow a closed form expression for the targeted bounding limit cycles in Section III and a tractable stability analysis in Section IV. The simplified massless-leg representation of a quadrupedal robot bounding in the sagittal-plane. ...
... Here, deadbeat control refers to a strategy resulting in exact correction to perturbations in a finite (typically minimum) number of steps[34]. ...
This paper develops a three degree-of-freedom sagittal-plane hybrid dynamical systems model of a bounding quadruped. Simple within-stance controls yield a closed form expression for a family of hybrid limit cycles that represent bounding behavior over a range of user-selected fore-aft speeds as a function of the model's kinematic and dynamical parameters. Controls acting on the hybrid transitions are structured so as to achieve a cascade composition of in-place bounding driving the fore-aft degree of freedom thereby decoupling the linearized dynamics of an approximation to the stride map. Careful selection of the feedback channels used to implement these controls affords infinitesimal deadbeat stability which is relatively robust against parameter mismatch. Experiments with a physical quadruped reasonably closely match the bounding behavior predicted by the hybrid limit cycle and its stable linearized approximation.
... In this paper, we consider the extended model in [15] since it was specifically proposed for energetically efficient control of locomotion on robotic platforms. Interested reader can see [16], [17], [18] for other extensions of the SLIP model. ...
... with I( f (t),t 0 ,t f ) and D( f (t),t 0 ,t f ) denoting first and second integrals of the function f (t) from t 0 to t f . Now, using (17) with t = t td , we obtaiṅ θ (t) =θ (0) + (l(0) −l(t))/I b + (∆x u ) I(R y (τ),t td ,t) θ (t) =θ (0) +θ (0)t + ∆L(t)/I b + (∆x u ) D(R y (τ),t td ,t). ...
A long-standing argument in model-based control of locomotion is about the level of complexity that a model should have to define a behavior such as running. Even though goldilocks model based on biomechanical evidence is often sought, it is unclear what complexity level qualifies to be such a model. This dilemma deepens further for bipedal robotic running with point feet, since these robots are underactuated, while tracking center-of-mass (COM) trajectories defined by the spring-loaded inverted pendulum (SLIP) model of running allocates all control inputs, leaving angular coordinates of the robot's trunk uncontrolled. Existing work in the literature approach this problem either by trading off COM trajectories against upright trunk posture during stance or by adopting more detailed models that include effects of trunk angular dynamics. In this paper, we present a new approach based on modifying foot placement targets of the SLIP model. Theoretical analysis and numerical results show that the proposed approach outperforms these traditional strategies.
... A large body of research in the SLIP literature has been directed towards more accurate and realistic controls, most of which may be categorized into two schemes: the methods which implement dead-beat like controllers through solving the running return maps [10]- [13]; and tabular control methods relying on look-up tables constructed upon the data generated by comprehensive forward-in-time simulations covering a wide range of SLIP states and parameters [14]- [16]. Application of the former to online control is not preferred, due to the non-linear optimization inevitably involved in the computations. ...
... Inside the controller, a quadratic programming (QP) problem is formulated. The optimization variable x = [q T , λ T ] T combines generalized acceleration and ground reaction forces, joint torques τ can be easily calculated with (13). The cost function is a weighted combination of multiple tasks: ...
To generate dynamic motions such as hopping and running on legged robots, model-based approaches are usually used to embed the well studied spring-loaded inverted pendulum (SLIP) model into the whole-body robot. In producing controlled "SLIPic" behaviors, existing methods either suffer from online incompatibility or resort to classical interpolations based on lookup tables. Alternatively, this paper presents the application of a data-driven approach which obviates the need for solving the inverse of the running return map online. Specifically, a deep neural network is trained offline with a large amount of simulation data based on the SLIP model to learn its dynamics. The trained network is applied online to generate reference foot placements for the humanoid robot. The references are then mapped to the whole-body model through a QP-based inverse dynamics controller. Simulation experiments on the WALK-MAN robot are conducted to evaluate the effectiveness of the proposed approach in generating bio-inspired and robust running motions.
... The fastest possible convergence is a one-step dead-beat stabilization in which perturbations are nullified in a single step [33]. Carver et al. [34] demonstrated that the number of steps needed for dead-beat stabilization depends on the number of goals (e.g., forward velocity and motion direction) and number of control actions. If there are n goals and m control actions such that m ≥ n, then it is possible to cancel the effect of perturbations in a single step. ...
Inspired by biological control synergies, wherein fixed groups of muscles are activated in a coordinated fashion to perform tasks in a stable way, we present an analogous control approach for the stabilization of legged robots and apply it to a model of running. Our approach is based on the step-to-step notion of stability, also known as orbital stability, using an orbital control Lyapunov function. We map both the robot state at a suitably chosen Poincaré section (an instant in the locomotion cycle such as the mid-flight phase) and control actions (e.g., foot placement angle, thrust force, braking force) at the current step, to the robot state at the Poincaré section at the next step. This map is used to find the control action that leads to a steady state (nominal) gait. Next, we define a quadratic Lyapunov function at the Poincaré section. For a range of initial conditions, we find control actions that would minimize an energy metric while ensuring that the Lyapunov function decays exponentially fast between successive steps. For the model of running, we find that the optimization reveals three distinct control synergies depending on the initial conditions: (1) foot placement angle is used when total energy is the same as that of the steady state (nominal) gait; (2) foot placement angle and thrust force are used when total energy is less than the nominal; and (3) foot placement angle and braking force are used when total energy is more than the nominal.
