Көп дәрежелі кинематикалық тізбектердің геометриялық талдауы

Geometric analysis of multi DoF kinematic chains that model a robot

In robotics, robot kinematics applies geometry to the study of the movement of multi degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation. Robot kinematics studies the relationship between the dimensions and connectivity of kinematic chains and the position, velocity and acceleration of each of the links in the robotic system, in order to plan and control movement and to compute actuator forces and torques. The relationship between mass and inertia properties, motion, and the associated forces and torques is studied as part of robot dynamics.

Forward kinematics specifies the joint parameters and computes the configuration of the chain. For serial manipulators this is achieved by direct substitution of the joint parameters into the forward kinematics equations for the serial chain. For parallel manipulators substitution of the joint parameters into the kinematics equations requires solution of the a set of polynomial constraints to determine the set of possible end effector locations.

Inverse kinematics specifies the end effector location and computes the associated joint angles. For serial manipulators this requires solution of a set of polynomials obtained from the kinematics equations and yields multiple configurations for the chain. The case of a general 6R serial manipulator (a serial chain with six revolute joints) yields sixteen different inverse kinematics solutions, which are solutions of a sixteenth degree polynomial. For parallel manipulators, the specification of the end effector location simplifies the kinematics equations, which yields formulas for the joint parameters.

The time derivative of the kinematics equations yields the Jacobian of the robot, which relates the joint rates to the linear and angular velocity of the end effector. The principle of virtual work shows that the Jacobian also provides a relationship between joint torques and the resultant force and torque applied by the end effector. Singular configurations of the robot are identified by studying its Jacobian.

The robot Jacobian results in a set of linear equations that relate the joint rates to the six vector formed from the angular and linear velocity of the end effector, known as a twist. Specifying the joint rates yields the end effector twist directly. The inverse velocity problem seeks the joint rates that provide a specified end effector twist. This is solved by inverting the Jacobian matrix. It can happen that the robot is in a configuration where the Jacobian does not have an inverse. These are termed singular configurations of the robot.

The principle of virtual work yields a set of linear equations that relate the resultant force torque six vector, called a wrench, that acts on the end effector to the joint torques of the robot. If the end effector wrench is known, then a direct calculation yields the joint torques. The inverse statics problem seeks the end effector wrench associated with a given set of joint torques, and requires the inverse of the Jacobian matrix. As in the case of inverse velocity analysis, at singular configurations this problem cannot be solved. However, near singularities small actuator torques result in a large end effector wrench. Thus near singularity configurations robots have large mechanical advantage.

Robot kinematics also deals with motion planning, singularity avoidance, redundancy, collision avoidance, as well as the kinematic synthesis of robots.