Dynamic Analysis of Manipulator Arm for 6-legged Robot

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Figure 2. Kinematic configuration of manipulator arm

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As can be seen from the Figure 1 the rotation motion of manipulator links is ensured by prismatic joints of hydraulic motors. Extension of motors is from 0 to 200 mm. Range of movement of particular joints is:

q1 <0°; 360°>, q2 <5°; 65°>, q3 <38°; 137°> and q4 <2°, 88°>.

2.2. Direct Kinematic Model

By direct kinematic model can be determined the end-effector position by knowing of all general variables q. By considering of particular transformation matrices (1 - 6) we obtain final transformation matrix (7) by product of these matrices. Final transformation matrix (7) determines position and orientation of end-effector in global coordinate system x₀, y₀, z₀.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Final transformation matrix (7) is squared matrix with dimension 4x4. The first, second and third element in last column of this matrix represent position of end-effector. By determining of direct kinematic model the workspace of manipulator arm can be plotted. For this purpose the suitable recursive algorithm in Matlab was written. The step of particular joint motion in this algorithm is 10° - 15°. For more precise plotting of workspace the step should decreases.

The workspace of manipulator arm in the Figure 3, Figure 4, Figure 5 is shown.

Figure 3. Workspace of manipulator arm

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Figure 4. Workspace of manipulator arm in X-Y axis

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Figure 5. Workspace of manipulator arm in X-Z axis

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From the Figure 4, Figure 5 it is obvious that range of motion from the axis of rotation around z to the last point of end-effector, in x-axis direction is roughly 3 m. It should be note that this range is so-called reachable workspace and it is only theoretical. The workspace is often broken down into a reachable workspace and a dexterous workspace. The reachable workspace is the entire set of points reachable by the manipulator, whereas the dexterous workspace consists of those points that the manipulator can reach with an arbitrary orientation of the end-effector. Obviously the dexterous workspace is a subset of the reachable workspace ^[3].

3. Trajectory Planning for Manipulator Arm

To achieve required motion of actuators there should be planned the trajectory of motion. For design the trajectory with required position, velocity and acceleration there has to be used polynomial of 5^th order. Let´s consider following equation ^[4].

(8)

By derivation according time we obtain polynomial for velocity

(9)

By next derivation we obtain polynomial for acceleration

(10)

By determination of initial and terminal conditions for position, velocity and acceleration we obtain 6 equations with 6 unknown constants a₀, a₁, a₂, a₃, a₄, a₅. Let´s consider for initial position the zero position, zero velocity and zero acceleration. Initial time is 0 s.

(11)

(12)

(13)

For terminal conditions let´s again consider zero velocity and zero acceleration. Final position is q_F.

(14)

(15)

(16)

Now, we have 6 equations with 6 unknown constants. Using Matlab the constants a₀, …, a₅ can be calculated. As final position is considered the maximal extension of hydraulic piston, what is 200 mm. Final time is chosen to be 10 seconds. The solution of equations (11) – (16) is

(17)

The course of position, velocity and acceleration of hydraulic piston in the Figure 6 is shown.

Figure 6. Course of position, velocity and acceleration of hydraulic piston

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Trajectory according the Figure 6 is trajectory for one of the hydraulic piston. For the second can be trajectory planned by similar way. It is logical, that the shorter time the piston will moves according planned path, the higher dynamic reactions will arise in this joint. From this reason the prismatic motion of hydraulic piston lasts 10 seconds, see Figure 6. The time of trajectory depends on requirements of performing operation.

Now, planned trajectory of particular hydraulic pistons will be used as input to the mechanic system.

4. Simulation of Manipulator Arm Motion Using Required Trajectory

The simulation manipulator arm motion in Matlab / SimMechanics - First Generation is done. SimMechanics is strong tool for investigation of mechanic or mechatronic systems. It significantly facilitates task of dynamic analysis especially in the case of redundant mechanisms. ^[5].

The input to joint actuator is required position, velocity and acceleration of hydraulic piston, which in the section 3 were determined. In this study we do not investigated behavior of manipulator using feedback control. The focus of study is just investigation of dynamic reactions affecting the particular links of manipulator. The task when position, velocity and acceleration are known and the torques have to be determined is task of inverse dynamics. Solution of inverse dynamics is determination of particular torques or forces, which are necessary to reach required position, velocity and acceleration. ^[2].

In the Figure 7 the simulation scheme from SimMechanics is shown.

Figure 7. SimMechanics scheme of manipulator arm

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In the study there was investigated motion of manipulator arm lifting the object with mass 200 kg.

The simulation process in the following figures is shown.

Figure 8. Course of simulation in SimMechanics a) 0s b) 2s c) 4s d) 6s e) 10s

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In the Figure 8 the sequence of motion can be seen. During the simulation each of manipulator link moves according to different trajectory.

Now, for our study it is important to know the reactions which affect manipulator arm link in the nodes CS1, CS2, CS3 and CS4, see Figure 9. Using block Joint Sensor the reaction in particular axis can be determined.

Figure 9. Manipulator link contact points of investigation

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In the following figures the course of reaction forces in the nodes CS1, CS2, CS3 and CS4 are shown.

Figure 10. Courses of reaction forces in chosen manipulator link nodes

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