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Abbreviations and Glossary Appendix A Abbreviations and Glossary Abbreviations are defined and the mathematical symbols and notations used in this book are specified. Furthermore, the random number generator used in this book is referenced. A.1 Abbreviations arccos arccosine BiRRT Bidirectional rapidly growing random tree C-space Configuration space DH Denavit-Hartenberg DLR German aerospace center DOF Degree of freedom FFT Fast fourier transformation IK Inverse kinematics HRI Human-Robot interface LWR Light weight robot MMI Institute of Man-Machine interaction PCA Principal component analysis PRM Probabilistic road map RRT Rapidly growing random tree rulaCapMap Rula-restricted capability map RULA Rapid upper limb assessment SFE Shape fit error TCP Tool center point OV workspace overlap 130 A Abbreviations and Glossary A.2 Mathematical Symbols C configuration space K(q) direct kinematics H set of all homogeneous matrices WR reachable workspace WD dexterous workspace WV versatile workspace F(R,x) function that maps to a homogeneous matrix VRobot voxel space for the robot arm VHuman voxel space for the human arm P set of points on the sphere Np set of point indices for the points on the sphere No set of orientation indices OS set of all homogeneous frames distributed on a sphere MS capability map A.3 Mathematical Notations a scalar value a vector aT vector transposed A matrix AT matrix transposed < a,b > inner product 3 SO(3) group of rotation matrices ∈ IR SO(3) := R ∈ IR 3×3| RRT = I,detR =+1 SE(3) IR 3 × SO(3) A TB reference frame B given in coordinates of reference frame A a ceiling function a floor function A.4 Random Sampling In this book, the drawing of random samples is often used. Unless otherwise stated, the samples are always drawn from an uniform distribution. The Mersenne Twister random number generator [73] is used to generate uniformly distributed random numbers. Appendix B Kinematics Descriptions This section describes homogeneous matrices and gives the specifications of arm kinematics used in this book in Denavit-Hartenberg (DH) parameters as proposed by Craig [19]. B.1 Homogeneous Matrices Once a base coordinate system A is chosen, any point in the 3D Cartesian space can be denoted by a 3x1 vector. T tA =(tx,ty,tz) (B.1) The orientation of a body in Cartesian space is described by a 3x3 rotation matrix R ∈ SO(3). One way to describe the body-attached coordinate system B, is to write the unit vectors of its principal axes in terms of the coordinate system A.Letx, y, z, be the unit vectors that give the principal directions of coordinate system B in terms of coordinate system A. If they are inserted as the columns of a 3x3matrix,a rotation matrix results. R = (xyz) (B.2) The information needed to completely specify the pose of a manipulator hand is a position and an orientation. The point on the manipulator hand whose position is described is chosen as the origin of the body-attached frame. A frame is a set of four vectors defining the position and orientation of an attached body in space. A homogeneous matrix is a 4x4matrix R t T = (B.3) 0T 1 used to describe a frame. A frame is a coordinate system where in addition to the orientation, a position vector is given which locates its origin relative to some other embedding frame [19]. Using frames the rigid body motion of objects in Cartesian space can be described. 132 B Kinematics Descriptions B.2 DLR LWR Kinematics In Table B.1 the kinematics is described using DH-parameters. The lower limits are shown in column ll and the upper limits in column ul. In Figure B.1 the individual positions and directions of the rotation axes are shown. The rotation axis of a link is always the z-axis shown as a blue arrow. Table B.1 DH parameters for the DLR LWR robot arm. Link lengths are in m. Angles are given in degrees. The robot base frame is identical with the frame of the first link. i αi−1 ai−1 di θi ll ul 0 0 0 0 0 -170 170 1 0 90 0 0 -120 120 2 0 -90 0.4 -90 -170 170 3 0 90 0 0 -120 120 4 0 -90 0.39 180 -170 170 5 0 90 0 90 -45 80 6 0 -90 0 90 -45 135 B.3 Kuka LWR Kinematics In Table B.2 the kinematics is described using DH-parameters. The lower limits are shown in column ll and the upper limits in column ul. In Figure B.2 the individual positions and directions of the rotation axes are shown. The rotation axis of a link is always the z-axis shown as a blue arrow. Table B.2 DH parameters for the Kuka LWR robot arm. Link lengths are in m. Angles are given in degrees. i αi−1 ai−1 di θi ll ul 0 0 0 0 0 -170 170 1 0 90 0 0 -120 120 2 0 -90 0.4 0 -170 170 3 0 -90 0 0 -120 120 4 0 90 0.39 0 -170 170 5 0 90 0 0 -130 130 6 0 -90 0 0 -170 170 B.4 Schunk PowerCube Arm Kinematics 133 Fig. B.1 Visualization of the kinematics of the DLR LWR robot arm that serves as the right arm for the humanoid robot Justin. The positions and directions of the link frames are shown. Each link rotates about its z-axis (blue arrow). A circle with a dot signifies that the respective axis is normal to the picture plane and is pointing at the observer. A circle with a cross signifies that the respective axis is normal to the picture plane and is pointing away from observer. B.4 Schunk PowerCube Arm Kinematics In Table B.3 the DH parameter and link limits are listed for the 6 DOF Schunk PowerCube arm. The kinematics is e.g. used in Section 4.4.3. The TCP used for the generation of the reachability sphere map is listed in the last row. 134 B Kinematics Descriptions Fig. B.2 Visualization of the kinematics of the Kuka LWR robot arm. The positions and directions of the link frames are shown. Each link rotates about its z-axis (blue arrow). Table B.3 DH parameters and link limits for the 6 DOF PowerCube arm. The DH parameters for the transformation to the robot arm TCP are shown in the last row of the table. Link lengths are in m. Angles are given in degrees. i ai−1 αi−1 d θi ll ul 1 0 0 0 0 -270 270 2 0 -90 0.26 -90 -270 270 3 0 -90 0 180 -135 135 4 0 -90 0.31 180 -270 270 5 0 -90 0 180 -120 120 6 0 -90 0 180 -270 270 TCP 0 0 0.265 0 B.6 Human Kinematics 135 B.5 Kuka Kr16 Kinematics In Table B.4, the DH-parameters are given for the industrial robot arm Kuka Kr16. It is significantly larger than the other robot arms used in this book. Fully extended it has a length of about 2 m. Table B.4 DH parameters and link limits for the 6 DOF Kuka Kr16 industrial robot arm. The DH parameters for the transformation from the last link to the robot arm TCP are shown in the last row of the table. Link lengths are in m. Angles are given in degrees. i ai−1 αi−1 d θi ll ul 1 0 180 -0.675 0 -185 185 2 0.26 90 0 0 -155 35 3 0.68 0 0 -90 -130 154 4 -0.035 90 -0.67 0 -350 350 5 0 -90 0 0 -130 130 6 0 -90 0.158 90 -350 350 TCP 0 0 0 0 B.6 Human Kinematics In Table B.5 the DH parameters are listed for the kinematics 2 that is examined in Section 5.2.2.2 as a candidate for the kinematics of the human right arm. Table B.5 DH parameters and link limits for the human arm kinematics 2. Link lengths are in m. Angles are given in degrees. i αi−1 ai−1 di θi ll ul 0 0 0 90 0 -50 180 1 0 -90 0 -90 -30 180 2 0 90 0.27 -90 -110 80 3 0 -90 0 0 0 145 4 0 90 0.22 0 -85 90 5 0 -90 0 90 -45 15 6 0 90 0 90 -85 85 136 B Kinematics Descriptions B.7 TCP Frames B.7.1 TCP Frames Used in Section 4.3 The TCP used in Section 4.3 for the DLR LWR arm which serves as the right arm for the robot Justin is given in DH-parameters in Equation B.4. The TCP frame is placed at the end of the robot arm. (αi−1,ai−1,di,θi)=(0,−90,0.118 m,−180) (B.4) The TCP used in Section 4.3 for the Kuka LWR arm is given in DH-parameters in Equation B.5. (αi−1,ai−1,di,θi)=(0,0,0.118 m,0) (B.5) The TCPs have the same pose with respect to the robot arm base to have arms with comparable length and comparable kinematics. Only if the TCPs are chosen like this can the capability maps directly be compared because in this case the robots try to reach the same poses.
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