We present an online trajectory optimization method and software platform applicable to complex humanoid robots performing challenging tasks such as getting up from an arbitrary pose on the ground and recovering from large disturbances using dexterous acrobatic maneuvers. The resulting behaviors, illustrated in the attached video, are computed only 7 × slower than real time, on a standard PC. The video also shows results on the acrobot problem, planar swimming and one-legged hopping. These simpler problems can already be solved in real time, without pre-computing anything.

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Synthesis and stabilization of complex behaviors through online trajectory optimization

Optimization is an appealing way to compute the motion of an animated character because it allows the user to specify the desired motion in a sparse, intuitive way. The difficulty of solving this problem for complex characters such as humans is due in part to the high dimensionality of the search space. The dimensionality is an artifact of the problem representation because most dynamic human behaviors are intrinsically low dimensional with, for example, legs and arms operating in a coordinated way. We describe a method that exploits this observation to create an optimization problem that is easier to solve. Our method utilizes an existing motion capture database to find a low-dimensional space that captures the properties of the desired behavior. We show that when the optimization problem is solved within this low-dimensional subspace, a sparse sketch can be used as an initial guess and full physics constraints can be enabled. We demonstrate the power of our approach with examples of forward, vertical, and turning jumps; with running and walking; and with several acrobatic flips.

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Synthesizing physically realistic human motion in low-dimensional, behavior-specific spaces

A new method for smooth trajectory planning of robot manipulators

Adaptive sliding mode control of electro-hydraulic system with nonlinear unknown parameters

A technique for optimal trajectory planning of robot manipulators is presented in this paper. In order to get the optimal trajectory, an objective function composed of two terms is minimized: a first term proportional to the total execution time and another one proportional to the integral of the squared jerk (defined as the derivative of the acceleration) along the trajectory. This latter term ensures that the resulting trajectory is smooth enough. The proposed technique enables one to take into account kinematic constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. Moreover, it does not require the total execution time of the trajectory to be set a priori. The algorithm has been tested in simulation yielding good results, also in comparison with those provided by another important trajectory planning technique.

A technique for time-jerk optimal planning of robot trajectories

This paper presents the derivation, simulation, and implementation of a nonlinear tracking control law for a hydraulic servosystem. An analysis of the nonlinear system equations is used in the derivation of a Lyapunov function that provides for exponentially stable force trajectory tracking. This control law is then extended to provide position tracking. The proposed controller is simulated and then implemented on an experimental hydraulic system to test the limits of its performance and the realistic effects of friction.

Experiments and simulations on the nonlinear control of a hydraulic servosystem

This paper presents an optimization-based framework for emulating the low-level capabilities of human motor coordination and learning. Our approach rests on the observation that in most biological motor learning scenarios some form of optimization with respect to a physical criterion is taking place. By appealing to techniques from the theory of Lie groups, we are able to formulate the equations of motion of complex multibody systems in such a way that the resulting optimization problems can be solved reliably and efficiently—the key lies in the ability to compute exact analytic gradients of the objective function without resorting to numerical approximations. The methodology is illustrated via a wide range of optimized, “natural” motions for robots performing various human-like tasks—for example, power lifting, diving, and gymnastics. © 2001 John Wiley & Sons, Inc.

Optimal Robot Motions for Physical Criteria

In this work an efficient dynamics algorithm is developed, which is applicable to a wide range of multibody systems, including underactuated systems, branched or tree-topology systems, robots, and walking machines. The dynamics algorithm is differentiated with respect to the input parameters in order to form sensitivity equations. The algorithm makes use of techniques and notation from the theory of Lie groups and Lie algebras, which is reviewed briefly. One of the strengths of our formulation is the ability to easily differentiate the dynamics algorithm with respect to parameters of interest. We demonstrate one important use of our dynamics and sensitivity algorithms by using them to solve difficult optimal control problems for underactuated systems. The algorithms in this paper have been implemented in a software package named Cstorm (Computer simulation tool for the optimization of robot manipulators), which runs from within Matlab and Simulink. It can be downloaded from the website http://www.eng.uci.edu/ bobrow/ @DOI: 10.1115/1.1376121#

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A Recursive Multibody Dynamics and Sensitivity Algorithm for Branched Kinematic Chains

Presents the derivation, simulation and implementation of a nonlinear tracking control law for a hydraulic servosystem. An analysis of the nonlinear system equations is used in the derivation of a Lyapunov function that provides for exponentially stable force trajectory tracking. This control law is then extended to provide position tracking. The proposed controller is simulated and then implemented on an experimental hydraulic system to test the limits of its performance and the realistic effects of friction.

In this paper we describe a technique to generate realistic human movement, specifically platform dives, by solving an optimal control problem requiring little a priori information. Solving the optimal control problem reliably requires computing exact analytic gradients of the objective function, which is made possible by a hybrid recursive algorithm that calculates the dynamics of the system. This algorithm is formulated with Lie algebra techniques and matrix exponentials, resulting in equations that are easily differentiable. This quality is essential when solving ill-conditioned systems such as the diver. Also, the importance of initial conditions in light of the constant angular momentum constraint is discussed.

G.A. Sohl

Papers

Experiments and simulations on the nonlinear control of a hydraulic servosystem

Optimal Robot Motions for Physical Criteria

A Recursive Multibody Dynamics and Sensitivity Algorithm for Branched Kinematic Chains

Experiments and simulations on the nonlinear control of a hydraulic servosystem

On the computation of optimal high-dives