In the previous video, we derived the product of exponentials formula to calculate T of theta, the configuration of the end-effector frame {b} relative to the fixed space frame {s}, when we're given the joint positions theta. In that formula, the joint screw axes are defined in the {s}-frame fixed to the world. In this video, we derive an alternative version of the formula where the joint screw axes are defined in the {b}-frame fixed to the end-effector. We use the same RPR robot as an example.

First let's move the robot to its zero configuration. As before, we define M to be the configuration of the {b}-frame when the robot is at its zero configuration.

Now we rotate joint 1 by an angle theta_1. The motion of the {b}-frame is a rotation about the screw axis of joint 1. We will represent the screw axis in the {b}-frame as B1, with the angular component omega1 and the linear component v1. Since the screw axis has rotation, omega_1 is a unit vector. Since the screw axis is aligned with the z-axis of the {b}-frame, omega1 is equal to zero, zero, one. The linear motion v1 can be obtained by visualizing a turntable at joint 1 rotating and measuring the linear velocity at a point at the origin of the {b}-frame. Since the distance between the joint 1 and the {b}-frame is 3, the linear velocity v_1 is zero, three, zero in the {b}-frame. We could also calculate this by defining a point q_1 on the axis of joint 1, where q_1 is expressed in the {b}-frame. Then v_1 is minus omega_1 cross q_1.

Now that we have the screw axis B_1, we can calculate the {b}-frame configuration T of theta. We simply apply the body-frame transformation corresponding to motion along the B_1 screw axis by an angle theta_1. This transformation is e to the bracket B_1 times theta_1. Since it is a body-frame transformation, it postmultiplies M.

Now suppose we change joint 2, extending it by theta2 units of distance. The screw axis B2 corresponding to joint 2 has zero angular component omega2, so the linear component v2 must be a unit vector. If we imagine the whole space translating at unit velocity along joint 2, a point at the origin of the {b}-frame would move with a linear velocity v2 equal to one, zero, zero, expressed in the {b}-frame. Therefore the screw axis B2 is defined as zero, zero, zero, one, zero, zero. The new configuration of the {b}-frame, T of theta, is obtained by right-multiplying the previous configuration by e to the bracket B_2 times theta_2.

Notice that the previous motion of joint 1 does not affect the relationship of joint 2's screw axis to the {b}-frame, because joint 1 is not between joint 2 and the {b}-frame. Therefore, B_2 is the same as the screw axis of joint 2 when the robot is at its zero configuration.

Finally, let's rotate joint 3 by theta_3. The screw axis B_3 is a pure rotation about an axis out of the screen, so the omega_3 vector is zero, zero, one. Rotation about this axis induces a linear motion v_3 equal to zero, one, zero in the {b}-frame. The new configuration of the {b}-frame, T of theta, is given by right-multiplying the previous configuration by the new body-frame transformation.

Again, the previous motions of joints 1 and 2 do not affect the relationship of joint 3's screw axis to the {b}-frame, because they are not between joint 3 and the {b}-frame. Therefore, B3 is the same as the screw axis of joint 3 when the robot is at its zero configuration.

In summary, we've derived a procedure for forward kinematics when the screw axes are expressed in the {b}-frame. First, define the M matrix representing the {b}-frame when the joint variables are zero. Second, define the {b}-frame screw axes B_1 to B_n for each of the n joint axes when the joint variables are zero. Finally, for the given joint values, evaluate the product of exponentials formula in the {b}-frame.

Comparing the two product of exponential formulas, in the {s}-frame and the {b}-frame, the major differences are the frame of representation of the screws and whether M is on the right side or the left side of the sequence of matrix multiplications.