3. Robotics Toolbox for Kinematics
The Robotic Toolbox in MATLAB is a comprehensive suite of functions and classes designed for simulating and analyzing the kinematics and dynamics of robotic manipulators. It provides tools for:
- Defining robot models: Using D-H parameters to specify the robot's geometry.
- Simulating movements: Visualizing and verifying kinematic models and trajectories.
- Solving kinematic equations: Both forward and inverse kinematics can be computed efficiently.
- Trajectory planning: Designing and optimizing paths for the manipulator to follow.
The Stanford manipulator consists five rotary joints and one prismatic (linear) joint. Each joint represents one degree of freedom. The kinematic parameters of the joints are θ1,θ2,d3,θ4,θ5,θ6.
The matrix which describes the end effector's position and attitude in reference coordinate system is expressed as follow:
$$ \left[\begin{array} {rrrr} nx & ox & ax & px \\ ny & oy & ay & py \\ nz & oz & az & pz \\ 0 & 0 & 0 & 1 \\ \end{array}\right] $$
Where n, o, a is the end effector's attitude, p is the end effector' position, px , py , pz is the end effector's coordinate in the datum reference coordinate system.
0 shows the relationship of the adjacent coordinate system n-1 and n. Firstly, the xn-1 axis is rotated by 0n angle around the zn-1 axis. Secondly, the xn-1 axis moves dn distance along the zn-1 axis. Thirdly, the zn-1 axis moves an distance along the rotating xn-1axis, that is, xn axis. Finally, the zn-1 axis is twisted by αn angle around the xn axis. As a result, the Matrix A is presented as follows:
$$A_n = Rz(θ_n)×T(a_n,d_n)×Rx(α_n) $$
$$ \left[\begin{array} {rrrr} \cos \theta_n & -\sin \theta_n & 0 & a_n \\ \sin \theta_n \cos \alpha_n & \cos \theta_n \cos \alpha_n & -\sin \alpha_n & -d_n \sin \alpha_n \\ \sin \theta_n \sin \alpha_n & \cos \theta_n \sin \alpha_n & \cos \alpha_n & d_n \cos \alpha_n \\ 0 & 0 & 0 & 1 \\ \end{array}\right] $$
$$ \left[\begin{array} {rrrr} \cos \theta_n & -\sin \theta_n & \sin \theta_n \sin \alpha_n & a_n \cos \alpha_n \\ \sin \theta_n & \cos \theta_n & -\cos \theta_n \sin \alpha_n & a_n \sin \theta_n \\ 0 & \sin \alpha_n & \cos \alpha_n & d_n \\ 0 & 0 & 0 & 1 \\ \end{array}\right] $$
Standard DH Parameters:
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