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The optimal motion planning problems for manipulator arms have been actively researched in robotics in the past two or three decades because the optimal motions that minimize actuator forces, energy consumption, or motion time yield high productivity, efficiency, smooth motion, durability of machine parts, etc. These merits are more apparent when the manipulator arms execute repeated motions. This subject is roughly divided into two categories according to the tasks that the manipulator arms should perform. These categories are characterized by motions as with or without geometric path constraints. If the geometric path of the end-effector of a non-redundant manipulator is predetermined, the motion has one degree of freedom (DOF) and can be represented by a scalar path variable. In this case, rigorous solutions were obtained subject to constant bounds on the actuator forces (Bobrow et al., 1985; Shin & McKay, 1985). Subsequently, the study was extended to cases where the path included certain singular points on it (Shiller, 1994) or where the actuator jerks and actuator torques were limited within constant bounds (Constantinescu & Croft, 2000). Most manipulator tasks—except arc welding, painting or cutting—are essentially motions without geometric path constraints. In this case, obstacle avoidance should be considered simultaneously with motion optimization. These types of manipulator motions have the same DOF as the number of lower pair joints, and the corresponding optimal motions are more complicated than those discussed above. The subject of this chapter lies in this category. Various types of methods have been developed to solve the optimal motion planning problem in the presence of obstacles. Optimal control theory (Bryson & Meier, 1990; Bessonnet & Lallemand, 1994; Formalsky, 1996; Galicki, 1998), nonlinear programming (Fenton et al., 1986; Bobrow, 1988; Singh & Leu, 1991), dynamic programming (Jouaneh et al., 1990), tessellation of joint (Ozaki & Lin, 1996) or configuration space (Shiller & Dubowsky, 1991), and a combination of these (Schlemmer & Gruebel, 1998; Hol et al., 2001) are the main techniques used. By the application of the optimal control theory, Pontryagin’s maximum principle leads to a two-point boundary value problem. Some researchers have attempted to solve these equations directly (Bryson & Meier, 1990; Formalsky, 1996) while others have attempted to

Optimal Motion Planning for Manipulator Arms Using Nonlinear Programming