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Cartesian coordinates are (x, y, 3 ), and its con guration space coordinates are ( 1 , 2 , 3 ) Typically, the user is interested in de ning the robot s paths in terms of Cartesian coordinates Robot s motors are controlled, however, in terms of joint values (that is, con guration space coordinates) Hence a standard task in robot motion planning and control is translation from one reference system to the other This process gives rise to two problems: (a) direct kinematics given joint values, nd the corresponding Cartesian coordinates and (b) inverse kinematics given the robot s Cartesian coordinates, nd the corresponding joint values As we will see in the next chapter, calculation of inverse kinematics is usually signi cantly more dif cult than the calculation of direct kinematics 12.

5 Motion Control The robot s path is a curve that the robot s end effector (or possibly its some other part) moves along in the robot workspace To be physically realizable, each point of the path must be associated with the joint values that fully describe the robot position and orientation in the respective con guration The term trajectory is used sometimes to designate a path geometry plus timing, velocity, and acceleration information along the path3 As used in robotics, the term motion control or motion control system refers to the lower-level control functions, such as algorithmic and electronic and mechanical means that direct individual motors, as opposed to motion planning, which signi es the upper-level control that is, control that requires some intelligence This is not to say that motion control is a simple matter robot controllers are often quite sophisticated.

A timing violation occurs when the data input to a ip- op transitions within a window around the active clock edge as de ned by the setup and hold times. This timing violation exists because if the setup and hold times are violated, a node within the ip- op (an internal node or one that is exposed to the outside world) can become suspended at a voltage that is not valid for either a logic-0 or logic-1. In other words, if the data is captured within the window described above, the transistors in the ip- op cannot be reliably set to a voltage representing logic-0 or logic-1. Rather than saturating at a high or low voltage, the transistors may dwell at an intermediate voltage before settling on a valid level (which may or may not be the correct level). This is called metastability and is illustrated in Figure 6.6.

The control means are used to realize a given path or trajectory with required delity While control means are beyond the scope of this book, for completeness we will review them brie y in the next chapter Depending on the number of DOF available for motion planning, we distinguish between three types of systems:.

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Holonomic Systems. These have enough DOF for an arbitrary motion. The minimum number of those is equal to the dimensionality of the corresponding C-space: For example, 6 is the minimum number of DOF a 3D arm manipulator needs to realize an arbitrary motion in space without obstacles. Nonholonomic Systems. These are systems with constraints on their motion. For example, a car is a nonholonomic system: with its 2-DOF control forward motion and steering it cannot execute a lateral motion; this creates a well-known dif culty in parallel car parking. Note that a car s C-space is 3D, with its axes being two position variables (x, y) plus the orientation angle. Redundant Systems. Those with the number of DOF well above the minimum necessary for holonomic motion. Humans, animals, and some complex robots present redundant systems.