Abstract
This paper introduces a local motion planning method for robotic systems with manipulating limbs, moving bases (legged or wheeled), and stance stability constraints arising from the presence of gravity. We formulate the problem of selecting local motions as a linearly constrained quadratic program (QP), that can be solved efficiently. The solution to this QP is a tuple of locally optimal joint velocities. By using these velocities to step towards a goal, both a path and an inverse-kinematic solution to the goal are obtained. This formulation can be used directly for real-time control, or as a local motion planner to connect waypoints. This method is particularly useful for high-degree-of-freedom mobile robotic systems, as the QP solution scales well with the number of joints. We also show how a number of practically important geometric constraints (collision avoidance, mechanism self-collision avoidance, gaze direction, etc.) can be readily incorporated into either the constraint or objective parts of the formulation. Additionally, motion of the base, a particular joint, or a particular link can be encouraged/discouraged as desired. We summarize the important kinematic variables of the formulation, including the stance Jacobian, the reach Jacobian, and a center of mass Jacobian. The method is easily extended to provide sparse solutions, where the fewest number of joints are moved, by iteration using Tibshirani’s method to accommodate an \(l_1\) regularizer. The approach is validated and demonstrated on SURROGATE, a mobile robot with a TALON base, a 7 DOF serial-revolute torso, and two 7 DOF modular arms developed at JPL/Caltech.
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Notes
- 1.
Refer to [16], Chap. 5 for background.
- 2.
These kinematics are well known, and arise from writing the no-slip conditions for each wheel with respect to the base frame.
- 3.
A desired velocity for the end-effector can be determined from the transformation between the current pose and the desired pose; the velocity (twist) corresponding to the error is determined by the matrix logarithm. See [16] Chap. 2.
- 4.
This is the coefficient matrix one obtains when the KKT conditions for this problem posed as a QP in standard form are written as a linear equation in primal and dual variables. See [17] for background.
- 5.
The worst case complexity of solving a QP with linear constraints is shown to be \(O(n^3 L)\) where \(n\) is the size of the decision variable, and \(L\) is the program input size [18].
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Shankar, K., Burdick, J.W., Hudson, N.H. (2015). A Quadratic Programming Approach to Quasi-Static Whole-Body Manipulation. In: Akin, H., Amato, N., Isler, V., van der Stappen, A. (eds) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-16595-0_32
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