M*g*L = TA ... (6) In order to estimate the torque required at each joint, we can must chose the worst case scenario.

Fig. 3 required torque at each joint

From the above figure we can see a link of required length L is rotate clockwise. Only  for perpendicular component of the length (L) between the pivot point and force (F) is taken into account. We can observe that the distance of length (L) is decreasing from length L3 to length L1. Since from the torque equation the length (L) or distance multiplied by the force (F), the greatest value will be obtained by using L3, The   force

(F) does not change. We can rotate the link counterclockwise similarly and observe the same effect.

The weight of the object (load) being held as Indicated in the Figure 4 by A1, which is multiplied  by  the distance its center of mass and the pivot point gives the torque required at the pivot. The tool takes into the consideration that the links may have a significant weight (W1, W2) and its center of mass is located at roughly the center of its length (L).

The torque caused by this difference masses must be added. The torque required at the first joint is therefore. T1 = L1*A1+L1*W1 (“A” is weight of the actuator or the load.) ................... (7)

We may consider that the actuator weight A2 which is as shown in the diagram below is not included when calculating the torque at that point. This is because by the length (L) between its center of mass and the pivot point is zero. The torque required at the 2nd joint must be re-calculated with the new lengths, which is as  shown in the following figure. (The applied torque shown in green color like T1 and T2)

T2 = L3*A1+ L1*W1+ L 2*A2+L4*W2     (8)

Knowing that the link weight (W2, W2) are located in the center point of the lengths, and the distance between the actuators (L1 and L3 shown in the diagram above) we can re-write the equation as follows:

T2 = (L1+L2)*A1+ (L1+L3)*W1+ (L 2)*A2+ (L2)*W2. (9)

Only for the tool requires that the user enter the lengths (L) of the each link, which would be L1 and L3 above so the equation is showing accordingly. The torques at each subsequent of the joint can be found similarly, by re-calculating the lengths between each new pivot point and each weight.

IV. THE KINEMATIC STRUCTURE OF BEGINNER ARM

The following figure shows the reference and link coordinate systems of 5- DOF arm using the first step of the animator. The values of the kinematic parameters are listed below in the table (I). Where lu, ls and lf are the link lengths of the shoulder, back arm and forearm respectively.

Fig. 5 The coordinate system of the arm

Defined a frame of each link, the coordinate transformation is describing the position and orientation of the end- effectors with respect to (wrt) the base frame is given by

Allude the basic construction of the forward kinematics function by composing the coordinate transformations into the one homogeneous transformation matrix. The actual description of the coordinate transformation system between frame i and frame I – 1 is given by homogeneous transformation matrix.

Where θ is denoted by the vector of the joint variables, a, b and c are the unit vectors of the frame attached to the end- effectors.

V. CONSIDERATION OF ELBOW POSITION

The main description is the intersection of the ellipsoid and the sphere mentioned above is analytically difficult. To make this task mathematically tractable, firstly we determine the intersection between the ellipsoid and the sphere at Z = ze. The redundancy curve of the arm can eventually be obtained by combining all such intersections for all values of ze in the interval. (Note that Zemin and Zemax are the minimal and the  maximal values of the Z coordinate of the elbow for a given position and orientation of the end-effectors.)

The intersection of the sphere with the plane Z = Ze results in a circle. Then the radius r2 and the center c2 of this circle can be determined as follows.