Kinematics and dynamics
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This chapter of the WEBook describes the mathematical models that represent the kinematics and dynamics of so-called “kinematic chains,” that is, the interconnection of a set of rigid links by means of ideal joints between the links.
Describing motion is a fundamental part of every robot control system. The simplest models for robot motion start from the idea that a robotic device is the interconnection of a set of rigid bodies (“links”) by means of kinematic pairs (“joints”). So, the motion of each link is constrained by its connection to the other links, and the exact form of the constraint is determined by (i) the type of the joints (most often revolute joints, sometimes also translational joints or spherical joints), and (ii) the relative position and orientation of the joint axes.
Kinematics is the study of how such an interconnected structure of links and joints can move, without taking into account what causes the motion, or what is the goal of the motion. Dynamics does take into account the forces that cause a motion, or that are the result of a motion. (And trajectory generation is the activity of planning the motion towards a given goal.) Dynamics describes the relationships between different time-based versions of motion on the one hand, and the motion generating force on the other hand:
- the inertia is the “mass distribution” within a robot system, and it defines the relationship between acceleration (i.e. the second order properties of motion) and force.
- the stiffness models the flexibility of the links and/or joints in the robot, and it defines the relationship between displacement (i.e. the zeroth order properties of motion) and force.
- the damping models the energy dissipation in the robot, and it defines the relationship between velocity (i.e. the first order properties of motion) and force.
Not surprisingly, kinematics and dynamics of robot systems rely to a very large extent on basic mathematics (geometry of position and orientation) and physics (Newton's laws). However, robotics does introduce some specific mathematics, in the form of the forward and inverse mappings (positions, motions, forces) between joint space and end-effector space. The complexity of these mappings depend mostly on the particular geometrical properties of the interconnection structure of a particular robot: most robots do not have an “arbitrary” structure, but a structure that drastically simplifies the above-mentioned mappings. Therefore the following categories of specific kinematic chain structures are discussed separately:
- Serial. The interconnection topology of the kinematic chains is a chain without loops or without multiple branches.
- Humanoid. The interconnection structure of the kinematic chain is a tree.
- Parallel. The interconnection structure of the kinematic chain consists of two "platforms" connected by a number of “legs” in parallel. These legs often have identical (mostly serial) kinematic structures.
- Mobile. The interconnection structure of the kinematic chain is that of two or more wheels connected to the same base. The base itself can, in turn, carry a kinematic chain too.
Another robotics-specific set of theory and practice, in the area of kinematics and dynamics, stems from the need to optimize relevant aspects of a robot system (speed, energy, dexterity, ...) when a particular motion task can be executed in more than one way. In this situation, the robot is called redundant for the given task. Redundancy can occur in several ways: the robot has more than six actuated joints, so every task for the end-effector can be done in more than one way; the task itself only requires less degrees of motion freedom than the robot can offer (e.g. spray painting or waterjet cutting).
The converse of redundancy also exists: the robot cannot execute the nominally desired motions because it puts more constraints on the robot motion than the available number of motion degrees of freedom. So, the robot controller has to compromise, and find the most optimal approximation of the desired motion. For example, the robot's end-effector is in contact with the environment, so some of its motions are constrained.
Here are some common mathematical concepts and models, for all sorts of kinematic chains.