An effector can be
![an effector can be an effector can be](https://myamazingthings.com/wp-content/uploads/2018/01/bedroom-plants-10-.jpg)
Recall that transformation matrices allow a given point to be transformed between different reference frames. However will we do it? Well, we know the distances from the shoulder to the elbow, and elbow to the wrist, as well as the joint angles, and we’re interested in finding out where the end-effector is relative to a base coordinate frame…OH MAYBE we should use those forward transformation matrices from the previous post. This says that not only does the Jacobian relate velocity from one state-space representation to another, it can also be used to calculate what the forces in joint space should be to effect a desired set of forces in end-effector space.įirst, we need to define the relationship between the position of the end-effector and the robot’s joint angles. Where is the Jacobian for the end-effector of the robot, and represents the forces in joint-space that affect movement of the hand. Setting these two equations (in end-effector and joint space) equal to each other and substituting in our equation for the Jacobian gives: Where is the torque applied to the joints, and is the angular velocity of the joints. Rewriting the above in terms of joint-space gives: Where is the force applied to the hand, and is the velocity of the hand. Rewriting this terms of end-effector space gives: Substituting in the equation for work into the equation for power gives:īecause of energy equivalence, work is performed at the same rate regardless of the characterization of the system. Power is the rate at which work is performed Where is work, is force, and is velocity. Work is the application of force over a distance Let the joint angle positions be denoted, and end-effector position be denoted. The planar two-link robot arm shown below will be used for illustration. How can we do this?Ĭonservation of energy is a property of all physical systems where the amount of energy expended is the same no matter how the system in question is being represented. As just mentioned, though, what we’re really interested in isn’t relating velocities, but forces. The above equivalence is a first step along the path to operational space control. Jacobians allow us a direct way to calculate what the control signal is in the space that we control (torques), given a control signal in one we don’t (end-effector forces).
![an effector can be an effector can be](http://docs.afterworks.com/FumeFX5C4D/Resources/Images/Doc/Effector_1.png)
Iin our robot arm, control is effected through a set of motors that apply torque to the joint angles, BUT what we’d like is to plan our trajectory in terms of end-effector position (and possibly orientation), generating control signals in terms of forces to apply in space. We’re interested in planning a trajectory in a different space than the one that we can control directly. Why is this important? Well, this goes back to our desire to control in operational (or task) space. This tells us that the end-effector velocity is equal to the Jacobian,, multiplied by the joint angle velocity. Where and represent the time derivatives of and. With a bit of manipulation we can get a neat result: You can think of a Jacobian as a transform matrix for velocity.įormally, a Jacobian is a set of partial differential equations: The Jacobian for this system relates how movement of the elements of causes movement of the elements of. For example, if we have a 2-link robotic arm, there are two obvious ways to describe its current position: 1) the end-effector position and orientation (which we will denote ), and 2) as the set of joint angles (which we will denote ). Basically, a Jacobian defines the dynamic relationship between two different representations of a system. Jacobian matrices are a super useful tool, and heavily used throughout robotics and control theory.