In two dimensions, two degrees of freedom will be required to apply an arbitrary force to the tip of the device, and one more if we want to control the torque also applied to the object. The gravity loading of the connections must also be compensated for and fortunately can be considered separately from the torque required to produce tip forces and torques. Note also that the geometry of the two-link device appears as a subproblem in many of the more complicated ones.
Given the two joint angles, let's calculate the location of the tip of the unit. When using a manipulator, one is more interested in calculating the joint angles that will place the tip of the device in a desired position. Note that these expressions are polynomials in the sines and cosines of the joint angles.
The set of points reachable from the tip of the device is a ring centered at the origin. So far we have thought of the manipulator as a device for placing the tip in any desired position within the workspace—that is, a position generator. Note that the forces applied to each of the pin nodes in the dev to balance as shown in the diagram.
Inspection of the equations shows that the resultant torques are linear in the applied forces, so we can use the superpower principle and calculate the torque induced by gravity separately.
14- DYNAMICS
In this connection, L is the Lagrangian or "kinetic potential", equal to the difference between the kinetic and potential energy, K - P. And again there is such an equation for each degree of freedom of the device. The other thing to remember is that the kinetic energy of a rigid body can be decomposed into a component due to the instantaneous linear translation of its center of mass (vmy2 ) and a component due to the instantaneous angular velocity (1702 ).
It will be useful to ignore gravity in the first round so that there will be no potential energy term. The moment of inertia for rotation about the center of mass of such a stick is (1/12)m12. To calculate the kinetic energy we need the linear and angular velocities of the links.
The same result could be obtained more directly by finding that the moment of inertia of the rod about one of its ends is (1/3)ml2.
This is partly due to the appearance of velocity-product terms, which represent centrifugal forces.
This, by the way, implies that if one makes the torque large enough to overcome the velocity-product terms, the links will move in the expected direction. In the diagram above, if a torque is applied to joint 1, the thrust exerted on the end of joint 2 is sufficient to cause it to have an angle. If we want to know exactly what accelerations will be produced by a given torque, we must solve for 01 and 82 in the above equations.
We can define the potential energy P as the sum of the products of the connecting masses and the height of their center of mass relative to an arbitrary plane. Because of the linearity of the equations, we can again use superposition and calculate the torque required to balance gravity separately. Similarly, it can be argued that not only should it be able to apply forces to the object, but also torque.
If we are restricted to operating in two-dimensional space, only one additional degree of freedom will be required, since it can only rotate about one axis, the axis normal to the plane of operation. It turns out that the same can be said for torque, as applying torque is easy. This is not much more difficult than it was for the two connections, since we can use the equation for 0 to eliminate one of the three joint angles from the other two equations, and so we only have to solve the two trigonometric polynomials again.
To determine how much of the workspace reachable by the manipulator can be used with arbitrary orientation of the last link, we could, as before, proceed with an algebraic approach. In cos(92) 141 and the recognition that the worst cases occur when the last term is parallel to the direction from the origin to. Not all points in the previously determined annular workspace can be reached with arbitrary orientation of the last link. One method of constructing the usable workspace is simply to construct a circle of radius 13 around each point.
A point is in the usable workspace if the circle thus constructed lies within the ring - previously defined. The width of the ring is twice the length of the shortest link minus 13, its average radius is still equal to the length of the longest link 1 and link 2. We have control over three torques T1, T2 and T3 and we want to use these to apply force F=(u,v) and torque T to the object held by the tip of the device.
We don't want to consider the movement of the manipulator right now, so imagine that its tip is firmly fixed in place. T We can therefore easily calculate what motor torques are needed to apply a given force and torque to the object.
The more general case involves much more arithmetic and the form of the end result is the same, only the numerical constant will be changed. Furthermore, we will ignore gravity for the time being and assume that the links are uniform rods with mass m and inertia (1/12)ml2 around their center of mass. Again we start by finding the rotational and translational speeds of each of the links. Apparently the angular velocities of the three links are eii (61+e2) and (1 +2 3.
There are few problems regarding position and force, since these quantities can be easily represented in n-dimensional space by n-dimensional vectors. It is incorrect to think of rotations "around an axis"; in our two-dimensional example such rotations would take one out of the plane of the paper, and in four dimensions not all possible rotations would be generated by considering only combinations of the four rotations about the coordinate axes. The coincidence that three variables are needed to specify a rotation in three dimensions allows some simplifications - for example, a torque can be calculated by taking cross products.
Corresponding to each node, then there will be a coordinate transformation from one system to another. In fact, the 3 x 3 rotation submatrix gives us the rotation of the end link with respect to the base and thus its orientation, while the 3 x 1 displacement submatrix is the position of the top of the end link with respect to the base. Given the shared variable, it is then a relatively straightforward matter to arrive at the position and orientation of the device or terminal tip.
One way to approach this problem would be to consider the 3 x 3 rotation submatrix consisting entirely of polynomials in the sines and cosines of the common angles and the 3 x I offset submatrix which also contains the length of connection and try to solve for the sines and cosines of the six angles of the connection. When solving polynomial equations by eliminating variables, the degree of the resulting polynomials increases with the product of the combined polynomials. Several such configurations are known, but one of the easiest to explain involves decoupling orientation from position.
Let's say the last three rotator cuffs intersect at one point, called the wrist. Then these last three can take care of the orientation, while the remaining three position the wrist, according to the orientation. Often the first three links are simply a combination of the two-link geometry we've already solved and the polar offset problem.
Now that we know the first three joint angles, we can calculate the orientation of the third that the wrist attaches to. The idea is to use the inherent compliance of the device as a kind of spring and drive the joints to angles slightly away from the equilibrium position. The dynamic behavior of a rigid body with respect to rotation can be conveniently expressed as a symmetric, square inertia matrJx.
