Fast Motions in Biomechanics and Robotics

An important technique for computing motions for robot systems is to conduct a numerical search for a trajectory that minimizes a physical criteria like energy, control effort, jerk, or time. In this paper, we provide example solutions of these types of optimal control problems, and develop a framework to solve these problems reliably. Our approach uses an efficient solver for both inverse and forward dynamics along with the sensitivity of these quantities used to compute gradients, and a reliable optimal control solver. We give an overview of our algorithms for these elements in this paper. The optimal control solver has been the primary focus of our recent work. This algorithm creates optimal motions in a numerically stable and efficient manner. Similar to sequential quadratic programming for solving finite-dimensional optimization problems, our approach solves the infinite-dimensional problem using a sequence of linear-quadratic optimal control subproblems. Each subproblem is solved efficiently and reliably using the Riccati differential equation.

A dynamic model of running-the spring-loaded inverted pendulum (SLIP)-has proven effective in describing the force patterns found in a wide variety of animals and in designing and constructing a number of terrestrial running robots. Climbing or vertical locomotion has, on the other hand, lacked such a simple and powerful model. Climbing robots to date have all been quasi-static in their operation. This paper introduces a one degree of freedom model of a climbing robot used to investigate the power constraints involved with climbing in a dynamic manner. Particular attention is paid to understanding how springs and body dynamics can be exploited to help relieve a limited power/weight ratio and achieve dynamic running and climbing.

In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control.

The objective of this study is to analyze the stability of two control strategies for a planar biped robot. The unexpected rotation of the supporting foot is avoided via the control of the center of pressure or CoP. For the simultaneous control of the joints and of the CoP, the system is under-actuated in the sense that the number of inputs is less than the number of outputs. Thus a control strategy developed for planar robot without actuated ankles can be used in this context. The control law is defined in such a way that only the geometric evolution of the biped configuration is controlled, but not the temporal evolution. The temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of a model with one degree of freedom. Simple conditions, which guarantee the existence of a cyclic motion and the convergence toward this motion, are deduced. These results are illustrated with some simulation results. In the first control strategy, the position of the CoP is tracked precisely, in the second one, only the limits on the CoP position are used to speed-up the convergence to the cyclic motion.

This paper first introduces a multi-locomotion robot with high mobility and then proposes Passive Dynamic Autonomous Control (PDAC) for the comprehensive control method of multiple types of locomotion. PDAC is the method to take advantage of the robot inherent dynamics and to realize natural dynamic motion. We apply PDAC to a biped walk control. On the assumption that the sagittal and lateral motion can be separated and controlled individually, each motion is designed based on the given desired step-length and period. In order to stabilize walking, the landing position control according to the status is designed. In addition, a coupling method between these motions, which makes the period of each motion identical, is proposed. Finally, we show that the multi-locomotion robot realizes the 3-dimensional dynamic walking using the PDAC control.

Movements of humans are achieved by muscle contractions. Humans are able to perform coordinated movements even in the presence of perturbations from the environment or of the muscles themselves. But which properties of the muscles and the geometry of the joints are responsible for the stability? Does the stability depend on the joint angle? How large are the perturbations, the muscle-skeletal system can cope with before reflexes or controls by the brain are necessary? To answer these questions, we will derive a mathematical model of the muscle-skeletal system without reflexes. We present different mathematical methods to analyze these systems with respect to the stability of movements and thus provide the mathematical tools to answer the above questions. This paper is a companion paper to [13] where the biological applications of the mathematical methods presented in this paper are discussed in more detail.

The Lateral Leg Spring model (LLS) was developed by Schmitt and Holmes to model the horizontal-plane dynamics of a running cockroach. The model captures several salient features of real insect locomotion, and demonstrates that horizontal plane locomotion can be passively stabilized by a well-tuned mechanical system, thus requiring minimal neural reflexes. We propose two enhancements to the LLS model. First, we derive the dynamical equations for a more flexible placement of the center of pressure (COP), which enables the model to capture the phase relationship between the body orientation and center-of-mass (COM) heading in a simpler manner than previously possible. Second, we propose a reduced LLS “plant model” and biologically inspired control law that enables the model to follow along a virtual wall, much like antenna-based wall following in cockroaches.

In the next few years considerable effort will be expended to make humanoid robots that can do true dynamic walking, or even running. One may numerically compute a desired gait, e.g. one that has been optimized to be asymptotically stable without feedback. One would normally give the gait as commands to the controllers for the robot joints. However, control system outputs generally differ from the command given, and the faster the command changes with time, the more deviation there is. Iterative learning control (ILC) and repetitive control (RC) aim to fix this problem in situations where a command is repeating or periodic. Since gaits are periodic motions, it is natural to ask whether ILC/RC can be of use in implementing gaits in hardware. These control concepts are no substitutes for feedback control but work in combination with them by adjusting the commands to the feedback controllers from a higher level perspective. It is shown that the gait problem does not precisely fit either the ILC or the RC problem statements. Gait problems are necessarily divided into phases defined by foot strike times, and furthermore the durations of the phases are not the same from cycle to cycle during the learning process. Several methods are suggested to address these issues, and four repetitive control laws are studied numerically. The laws that include both position and velocity error in the updates are seen to be the most effective. It appears that with appropriate refinement, such generalized RC laws could be very helpful in getting hardware to execute desired gaits.

Dynamic walking with two-legged robots is still an unsolved problem of todays robotics research. Beside finding mathematical models for the walking process, suitable mechanical designs and control methods must be found. This paper presents concepts for the latter two points. As biological walking makes use of the elastic properties of e.g. tendons and muscles, a joint design using a pneumatic rotational spring with adjustable stiffness is proposed. Equations to model the spring’s dynamics as well as the supporting sensor systems and electronics are presented. For controlling the robot a behaviour-based approach is suggested.

This paper deals with a methodology to design optimal reference trajectories for walking gaits. This methodology consists of two steps: (i) design a parameterized family of motions, and (ii) determine the optimal parameters giving the motion that minimizes a criterion and satisfies some constraints within this family. This approach is applied to a five link biped, the prototype Rabbit. It has point feet and four actuators which are located in each knee and haunch. Rabbit is underactuated in single support since it has no actuated feet and is overactuated in double support. To take into account this under-actuation, a characteristic of the family of motions considered is that the four actuated joints are prescribed as polynomials in function of the absolute orientation of the stance ankle. There is no impact. The chosen criterion is the integral of the square of torques. Different technological and physical constraints are taken into account to obtain a walking motion. Optimal process is solved considering an order of treatment of constraints, according to their importance on the feasibility of the walking gait. Numerical simulations of walking gaits are presented to illustrate this methodology.

For fast motions in biomechanics and robotics, stability and robustness against perturbations are critical issues. The faster a motion the more important it is to exploit the system’s natural stability properties for control. The stability of a periodic motion can be measured in terms of the spectral radius of the monodromy matrix. We optimize this stability criterion for a given robot topology, using special purpose optimization methods and leaving the model parameters, actuator inputs, trajectory start values and cycle time free to be determined by the optimization. This approach allows us to create simulations of robots that can move stably without any feedback. In order to analyze the robustness of a resulting periodic motion, we propose two methods, the first of which relies on forward simulations using perturbed start data and parameters while the second is based on the pseudospectra of the matrix. As a new example for a fast open-loop stable motion that has been produced by stability optimization, we present a biped gymnastics robot performing repetitive flip-flops (i.e. back handsprings). A similar model has previously been shown capable of performing open-loop stable running motions and repetitive somersaults.

This paper develops a class of bipedal running controllers based on the hybrid zero dynamics (HZD) framework and discusses related experiments conducted in September 2004 in Grenoble, France. In these experiments, RABBIT, a five-link, four-actuator, planar bipedal robot, executed six consecutive running steps. The observed gait was remarkably human-like, having long stride lengths (approx. 50 cm or 36% of body length), flight phases of significant duration (approx. 100 ms or 25% of step duration), an upright posture, and an average forward rate of 0.6 m/s. A video is available at [7, 17]. In the time allotted for experiments, stability of the gait could not be validated. To put the results into context, background information on hybrid robot modeling, control philosophy, and gait optimization techniques accompany final experimental observations. An additional discussion about some unmodeled dynamic and geometric effects that contributed to implementation difficulties is given.

We present velocity-based stability margins for fast bipedal walking that are sufficient conditions for stability, allow comparison between different walking algorithms, are measurable and computable, and are meaningful. While not completely necessary conditions, they are tighter necessary conditions than several previously proposed stability margins. The stability margins we present take into consideration a biped’s Center of Mass position and velocity, the reachable region of its swing leg, the time required to swing its swing leg, and the amount of internal angular momentum available for capturing balance. They predict the opportunity for the biped to place its swing leg in such a way that it can continue walking without falling down. We present methods for estimating these stability margins by using simple models of walking such as an inverted pendulum model and the Linear Inverted Pendulum model. We show that by considering the Center of Mass location with respect to the Center of Pressure on the foot, these estimates are easily computable. Finally, we show through simulation experiments on a 12 degree-of-freedom distributed-mass lower-body biped that these estimates are useful for analyzing and controlling bipedal walking.

The paper considers the use of sum of squares techniques in nonlinear model predictive control. To be more precise, sum of squares techniques are used to solve at each sampling instant a finite horizon optimal control problem which arises in nonlinear model predictive control for discrete time polynomial systems. The combination of nonlinear model predictive control and sum of squares techniques is motivated by the successful application of semidefinite programming in linear model predictive control. The advantages and disadvantages of applying sum of squares techniques to nonlinear model predictive control are illustrated on a small example.

Stability of biped walking is an important characteristic of legged locomotion. Whereas clinical investigations often relate increased variability to decreased stability, there are only few studies examining stability aspects directly. On the other hand, various papers from the field of robotics are dedicated to the question: how can the stability of legged locomotor systems be quantified? Particularly, when it comes to realizing fast motions in robots, the question of maintaining dynamic stability is of utmost importance. The current paper presents a theoretical comparison of several measures for dynamic stability — namely Floquet multipliers and Local Divergence Exponents (LDE). The sensitivity of these parameters to changes in speed of human treadmill locomotion is investigated. Experimental results show that two different types of stability with respect to speed dependence seem to exist. Short term LDE and Floquet multipliers consider the stability over a period of one stride, which seems to be optimal at intermediate walking speeds. Long term LDE quantify stability of movement trajectories over multiple strides. This type of stability decreases with speed and may be one reason for changing gaits from walking to running at a certain speed value.

The spring loaded inverted pendulum model (SLIP) has been shown to accurately model sagittal plane locomotion for a variety of legged animals. Tuned appropriately, the model exhibits passively stable periodic gaits using either fixed leg touchdown angle or swing-leg retraction protocols. In this work, we investigate the relevance of the model in insect locomotion and develop a simple feedback control law to enlarge the basin of stability and produce stable periodic gaits for both the point mass and rigid body models. Control is applied once per stance phase through appropriate choice of the leg touchdown angle. The control law is unique in that stabilization is achieved solely through direct observation of the leg angle and body orientation, rather than through feedback of system positions, velocities, and orientation.

It has long been the dream to build robots which could walk and run with ease. To date, the stance phase of walking robots has been characterized by the use of either straight, rigid legs, as is the case of passive walkers, or by the use of articulated, kinematically-driven legs. In contrast, the design of most hopping or running robots is based on compliant legs which exhibit quite natural behavior during locomotion.

Mechanical properties of complex biological systems are non-linear, e.g. the force-velocity-length relation of muscles, activation dynamics, and the geometric arrangement of antagonistic pair of muscles. The control of such systems is a highly demanding task. Therefore, the question arises whether these mechanical properties of a muscle-skeletal system itself are able to support or guarantee for the stability of a desired movement, indicating self-stability. Self-stability of single joint biological systems were studied based on eigenvalues of the equation of motions and the basins of attraction were analysed using Lyapunov functions. In general, we found selfstability in single muscle contractions (e.g. frog, rat, cui), in human arm and leg movements, the human spine and even in the co-ordination of complex movements such as tennis or basketball. It seems that self-stability may be a general design criterion not only for the mechanical properties of biological systems but also for motor control.

Walking, running or jumping are special cases of articulated motions which rely heavily on contact forces for their accomplishment. This central role of the contact forces is widely recognized now, but it is rarely connected to the structure of the dynamics of articulated motion. Indeed, this dynamics is generally considered as a complex nonlinear black-box without any specific structure, or its structure is only partly uncovered. We propose here to precise this structure and show in details how it shapes the movements that an articulated system might realize. Some propositions are made then to improve the design of control laws for walking, running, jumping or free-floating motions.

In human walking, the swing leg moves backward just prior to ground contact, i.e. the relative angle between the thighs is decreasing. We hypothesize that this swing leg retraction may have a positive effect on gait stability, because similar effects have been reported in passive dynamic walking models, in running models, and in robot juggling. For this study, we use a simple inverted pendulum model for the stance leg. The swing leg is assumed to accurately follow a time-based trajectory. The model walks down a shallow slope for energy input which is balanced by the impact losses at heel strike. With this model we show that a mild retraction speed indeed improves stability, while gaits without a retraction phase (the swing leg keeps moving forward) are consistently unstable. By walking with shorter steps or on a steeper slope, the range of stable retraction speeds increases, suggesting a better robustness. An optimization of the swing leg trajectory of a more realistic model also consistently comes up with a retraction phase, and indeed our prototype demonstrates a retraction phase as well. The conclusions of this paper are twofold; (1) use a mild swing leg retraction speed for better stability, and (2) walking faster is easier.