Part I

Kinematics

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Kinematics is the science of geometry in motion. It is restricted to apure geometrical description of motion by means of position, orientation,and their time derivatives. In robotics, the kinematic descriptions of ma-nipulators and their assigned tasks are utilized to set up the fundamentalequations for dynamics and control.Because the links and arms in a robotic system are modeled as rigid

bodies, the properties of rigid body displacement takes a central place inrobotics. Vector and matrix algebra are utilized to develop a systematic andgeneralized approach to describe and represent the location of the arms ofa robot with respect to a global fixed reference frame G. Since the arms ofa robot may rotate or translate with respect to each other, body-attachedcoordinate frames A,B,C, · · · or B1, B2, B3, · · · will be established alongwith the joint axis for each link to find their relative configurations, andwithin the reference frame G. The position of one link B relative to anotherlink A is defined kinematically by a coordinate transformation ATB betweenreference frames attached to the link.The direct kinematics problem is reduced to finding a transformation

matrix GTB that relates the body attached local coordinate frame B tothe global reference coordinate frame G. A 3 × 3 rotation matrix is uti-lized to describe the rotational operations of the local frame with respectto the global frame. The homogeneous coordinates are then introduced torepresent position vectors and directional vectors in a three dimensionalspace. The rotation matrices are expanded to 4× 4 homogeneous transfor-mation matrices to include both the rotational and translational motions.Homogeneous matrices that express the relative rigid links of a robot aremade by a special set of rules and are called Denavit-Hartenberg matricesafter Denavit and Hartenberg (1955). The advantage of using the Denavit-Hartenberg matrix is its algorithmic universality in deriving the kinematicequation of a robot link.The analytical description of displacement of a rigid body is based on

the notion that all points in a rigid body must retain their original relativepositions regardless of the new position and orientation of the body. Thetotal rigid body displacement can always be reduced to the sum of its twobasic components: the translation displacement of an arbitrary referencepoint fixed in the rigid body plus the unique rotation of the body about aline through that point.Study of displacement motion of rigid bodies leads to the relation be-

tween the time rate of change of a vector in a global frame and the timerate of change of the same vector in a local frame.Transformation from a local coordinate frame B to a global coordinate

frame G is expressed byGr = GRB

Br+GdB

where Br is the position vector of a point in B, Gr is the position vectorof the same point expressed in G, and d is the position vector of the origin

KinematicsPart I :

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o of the body coordinate frame B(oxyz) with respect to the origin O ofthe global coordinate frame G(OXY Z). Therefore, a transformation hastwo parts: a translation d that brings the origin o on the origin O, plus arotation, GRB that brings the axes of oxyz on the corresponding axes ofOXY Z.The transformation formula Gr = GRB

Br+GdB , can be expanded toconnect more than two coordinate frames. The combination formula fora transformation from a local coordinate frame B1 to another coordinateframe B2 followed by a transformation from B2 to the global coordinate Gis

Gr =¡GRB2

B2r+GdB2

¢+¡B2RB1

B1r+B2dB1

¢= GRB2

B2RB1

B1r+ GdB2 +B2dB1

= GRB1

B1r+GdB1 .

A robot consists of n rigid links with relative motions. The link attachedto the ground is link (0) and the link attached to the final moving link,the end-effector, is link (n). There are two important problems in kine-matic analysis of robots: the forward kinematics problem and the inversekinematics problem.In forward kinematics, the problem is that the position vector of a point

P is in the coordinate frame Bn attached to the end-effector as nrp, andwe are looking for the position of P in the base frame B0 shown by 0rp.The forward kinematics problem is equivalent to having the values of jointsvariable and asking for the position of the end-effector.In inverse kinematics, the problem is that we have the position vector of

a point P in the base coordinate frame B0 as 0rp, and we are looking fornrp, the position of P in the base frame B0. This problem is equivalent tohaving the position of the end-effector and asking for a set of joint variablesthat make the robot reach the point P .In this part, we develop the transformation formula to move the kine-

matic information back and forth from a coordinate frame to another co-ordinate frame.

KinematicsPart I :