Comparing Locally Energy-Optimal Newton Algorithms to Steer Driftless Systems
In this chapter two locally energy-optimal motion planning methods were compared. Both of them are based on the Newton algorithm (primarily exploited to solve an inverse kinematic task in robotic manipulators) adapted to motion planning of driftless nonho
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Abstract In this chapter two locally energy-optimal motion planning methods were compared. Both of them are based on the Newton algorithm (primarily exploited to solve an inverse kinematic task in robotic manipulators) adapted to motion planning of driftless nonholonomic systems. The first, continuous method employs a standard technique of optimization in the null-space of a Jacobian matrix. The matrix is derived from a linearization of the system along a current trajectory and the trajectory corresponds to a finite Fourier series representation of controls. The second, discrete one, admits only piecewise-constant controls modified at a finite number of sub-intervals of the time horizon. Some simulations performed on a model of the free-floating planar double pendulum placed atop of a base revealed that the continuous method is more reliable, although, when the goal configuration does not need to be reached very accurately, the discrete one can be useful and energy-effective. A classification of the discrete methods was also proposed. According to the classification the proposed discrete method is of the 1st order.
1 Introduction Many models of contemporary robots (mobile platforms and robots, free-floating robots, under actuated manipulators), considered at a kinematic level, belong to a class of driftless nonholonomic systems described by the equations q_ ¼
m X
gi ðqÞui
ð1Þ
i¼1
I. Duleba (&) J. Jakubiak M. Opalka Institute of Computer Engineering, Control and Robotics, Wroclaw University of Technology, Janiszewskiego Street 11/17 50-372 Wroclaw, Poland e-mail: [email protected]
J. Sa˛siadek (ed.), Aerospace Robotics, GeoPlanet: Earth and Planetary Sciences, DOI: 10.1007/978-3-642-34020-8_2, Springer-Verlag Berlin Heidelberg 2013
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where q = (q1, …, qn)T is a configuration vector, u = (u1, …, um)T are control inputs, and g1, g2, …, gm are smooth vector fields called generators of the system (1). A characteristic feature of the models (1) is that the number of state variables n is larger than the number of controls m. A motion planning task is to find controls u(t) = (u1(t), …, um(t))T that steer the system (1) from a given initial configuration q0 to a final one qf. Despite the fact that the condition m \ n makes this task difficult, still there are many methods to solve it (Duleba 1998; LaValle 2006). The most effective were designed for special subclasses of driftless systems [flat (Rouchon et al. 1993), nilpotent (Lafferriere and Sussmann 1991), in a chain form (Murray and Sastry 1993)]. However, still there is a need for general-purpose methods, especially equipped with some additional features (optimality, collision avoidance). In this chapter one of the methods will be considered, namely the Newton algorithm. Originally, it was designed to solve inverse kinematic tasks in robot manipulators, then extended to driftless nonholonomic systems (Divelbiss and Wen 1993; Duleba and Sasiadek 2003) by appropriately redefined kinematics (Tchon and Jakubiak 2003). The main i
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