Robots that interact with the environment are constrained dynamical systems. Constrained dynamical systems can be modeled using a set of differential equations subject to holonomic and non-holonomic constraints; algebraic constraints. One way to approximately model the constrained dynamics of these robots, for example walking robots, is to use switching-differential-algebraic equations (SDAEs), defined by a set of differential equations and a switching set of algebraic constraints. This idea suggests approximating unilaterally constrained non-holonomic dynamical systems with piecewise holonomic systems. We work on numerical simulators that can predict robot motion by efficiently solving differential-algebraic equations.
Modeling and Simulating Robot Motion using switching DAEs
We present a differential-algebraic formulation with switching constraints to model the non-smooth dynamics of robotic systems subject to changing constraints and multiple impacts. The formulation combines a single structurally simple governing equation, a set of switching kinematic constraints, and the plastic impact law, to represent the dynamics of robots that interact with their environment.
- Y. Li, H. Yu, and D.J. Braun, Algorithmic Resolution of Multiple Impacts in Non-smooth Mechanical Systems with Switching Constraints, IEEE International Conference on Robotics and Automation, pp. 7639-7645, Montreal, QC, Canada, 2019.
Modeling and Simulating Robot Motion using DAEs
A large class of constrained dynamical systems can be modeled using differential algebraic equations (DAEs). Unlike ordinary differential equations (ODEs), which can be accurately integrated with explicit numerical methods (for example the Runge-Kutta method), accurate numerical integration of differential-algebraic equations requires sophisticated implicit integration methods. In this research, we develop explicit numerical integration algorithms which are easy to implement and can be used to accurately integrate differential algebraic equations. We assumed that the numerical solution is error contaminated, such that, neither the kinematic constraints nor any type of motion invariant, for example, an energy conservation law, can be exactly satisfied by a numerical solution. Using this assumption, we perform systematic mathematical derivations to find novel numerical error correction terms for accurate numerical integration of the differential-algebraic equations of motion.
- D.J. Braun and M. Goldfarb, Simulation of Constrained Mechanical Systems – Part I: An Equation of Motion, ASME Journal of Applied Mechanics, vol. 79, issue 4, 041017, 2012.
- D.J. Braun and M. Goldfarb, Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration, ASME Journal of Applied Mechanics, vol. 79, issue 4, 041018, 2012.
- D.J. Braun, M. Goldfarb, Elimination of Constrained Drift in the Numerical Simulation of Constrained Dynamical Systems, Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 37-40, pp. 3151-3160, 2009.
Our method is easy to implement and provides significant improvement in numerical accuracy when tested against the analytical solution of simple case-study problems or the numerical solution of complex problems modeled using ordinary differential equations. The test example below demonstrates the relation between the integrator we developed to precisely solve differential algebraic equations and the Runge-Kutta method which can only be used to precisely solve differential equations.
