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Figure 2.1 A planar two-link arm manipulator: l1 and l2 are links, with their respective endpoints a and b; J0 and J1 are two revolute joints; 1 and 2 are joint angles. Both links are of the same thickness 2R. The robot s base coincides with the joint J0 and is xed.
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Position of the endpoint b can be de ned in the arm workspace either by two coordinates x and y in the Cartesian plane (x, y) or by two joint angles 1 and 1 . Hence we distinguish two representations of an arm con guration, in Cartesian space (x, y) and in joint space ( 1 , 1 ). An arm with more degrees of freedom or a three-dimensional arm will result in a higher complexity of those representations. 2.1 KINEMATICS Kinematics describes the relationship between positions, velocities, and accelerations of a set of bodies in our case, of the robot arm links. While we are at it, let us also de ne the concepts of statics and dynamics, which often go together with kinematics when describing a body s motion, and which we will address in the following sections: Statics describes (a) the relationship between forces and torques that, say, an arm manipulator exerts on the surrounding objects and (b) the relationship between internal forces and torques at the arm links. Dynamics describes the relationship between kinematics and statics. For example, the relationship between torques at the arm joints and link positions represents the arm s dynamics. For trajectory planning of robot arm manipulators, kinematics is especially important. Here is one reason for that. More often than not, people prefer to command arm s positions in terms of Cartesian coordinates in our case the two coordinates (x, y) whereas the arm control system expects them in terms of arm s joint values in our case the angular joint values ( 1 , 1 ). Inversely, if
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the arm somehow say, acting upon the sensor data arrives at some position, from the arm s joints we obtain its joint angles, and we would like to know which position (x, y) in Cartesian space they correspond to (Figure 2.1). Hence there is a need to translate from one coordinate system to the other. Accordingly, there are two relationships between these two coordinate representations: Direct Kinematics. Given the values ( 1 , 1 ), nd the corresponding Cartesian coordinates (x, y) of the arm endpoint. Inverse Kinematics. Given Cartesian coordinates (x, y) of the arm endpoint, nd the corresponding joint values ( 1 , 1 ). Note that if pi is the vector from the proximal to the distal joint of link i (Figure 2.2), i = 1, 2, then
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(2.1)
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Direct Transformation (Direct Kinematics). From Figure 2.2, it is not hard to derive equations for the joint position, and by taking their derivatives to nd equations for velocity and accelerations of the arm endpoint in terms of the arm joint angles:
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Position: X= l1 cos 1 + l2 cos( 1 + 2 ) x = y l1 sin 1 + l2 sin( 1 + 2 ) (2.2)
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A sketch for deriving the two-link arm s kinematic transformations.
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Velocity: l1 sin 1 l2 sin( 1 + 2 ) l2 sin( 1 + 2 ) x X= = y l1 cos 1 + l2 cos( 1 + 2 ) l2 cos( 1 + 2 ) 1 2 (2.3)
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or, in vector form, X = J , where the 2 2 matrix J is called the system s Jacobian (see, e.g., Refs. 6 and 7). Acceleration: l1 sin 1 l2 sin( 1 + 2 ) x = y l1 cos 1 l2 cos( 1 + 2 ) l1 cos 1 l2 cos( 1 + 2 ) l1 sin 1 l2 sin( 1 + 2 ) 1 1 + 2
2 1
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(2.4)
Inverse Transformation (Inverse Kinematics). From Figure 2.2, obtain the position and velocity of the arm joints as a function of the arm endpoint Cartesian coordinates: