by Dr. ir. Volkert van der Wijk, PhD (2014)
A mechanism or robotic manipulator that is dynamically balanced does not cause base vibrations when moving at high speeds. A balanced mechanism does not perturb equipment mounted on the same base, the floor, and machines placed nearby, therefore dynamic balance improves the accuracy of a machine, reduces cycle times, and reduces the required size, mass, and stiffness of base frames and floors. For robotic end-effectors dynamic balance is advantageous because of dynamic decoupling. A balanced mechanism also remains stationary in any position which is advantageous for realizing safe and energy efficient large motion of objects.
Since commonly dynamic balancing of multi-degree-of-freedom mechanisms is considered after the kinematic synthesis of a mechanism, profitable solutions without excessive mass, inertia, and complexity addition are hardly found. In this work a methodology was developed that considers dynamic balance as a design principle in the synthesis of dynamically balanced mechanisms. New mechanisms were found where all elements contribute to the motion as well as to the dynamic balance. Such mechanisms were named inherently dynamically balanced mechanisms.
Two methods for the synthesis of inherently dynamically balanced mechanisms were proposed which consider dynamic balance prior to the kinematic synthesis, the ‘method of linearly independent linear momentum’ and the ‘method of principal vector linkages’. Also a method was found by which the loop closure relations of general planar closed kinematic chains can be considered implicitly. For that purpose the mass of an element with general center-of-mass is modeled with one virtual equivalent mass and two real equivalent masses. For the first time a high-speed inherently dynamically balanced parallel manipulator was designed, built, and tested, showing that with dynamic balance the performance of the manipulator can be improved significantly. New multi-degree-of-freedom balanced kinematic mechanism solutions were synthesized for various applications such as a balanced grasper, a balanced bascule bridge without counter-mass, and balanced movable architecture.
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Principles of shaking force and shaking moment balance . . . . . . . . . . 1
1.2 Applications of dynamic balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Limitations of current balancing methods and balance solutions
for multi-degree-of-freedom mechanisms . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Focus on inherently balanced mechanism design . . . . . . . . . . . . . . . . 12
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Inherent force balance of given mechanisms with linear momentum. . 17
2.1 Open kinematic chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Closed kinematic chains with open chain method . . . . . . . . . . . . . . . . 21
2.2.1 4R four-bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Crank-slider mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Delta robot manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Closed kinematic chains including loop-closure relations . . . . . . . . . 26
2.3.1 4R four-bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Parallelogram and pantograph linkage . . . . . . . . . . . . . . . . . . . 34
2.3.3 4-RRR parallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.4 3-RRR parallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.5 2-RRR parallel manipulator
(6R six-bar mechanism and 5R five-bar mechanism) . . . . . . 49
2.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Principal vector linkages for inherent shaking force balance . . . . . . . . 53
3.1 The 2-DoF pantograph linkage as a principal vector linkage . . . . . . . 53
3.2 Principal vector linkage of three principal elements in series . . . . . . . 58
3.2.1 Union of pantographs and Fischer’s linkage . . . . . . . . . . . . . . 59
3.2.2 Generalization and calculation of principal dimensions
with Equivalent Linear Momentum Systems . . . . . . . . . . . . . . 61
3.2.3 Method of rotations about the principal joints . . . . . . . . . . . . . 72
3.2.4 Kinematic variations of the principal vector linkage . . . . . . . 72
3.3 Principal vector linkage of four principal elements in series . . . . . . . 77
3.4 Principal vector linkage of four principal elements in parallel . . . . . . 85
3.5 The spatial principal vector linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Closed-chain principal vector linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Approaches for synthesis of closed-chain principal vector linkages . 101
4.2 Closed chain of four elements with Open Chain Method . . . . . . . . . . 103
4.3 Mass equivalent model of a general element in a closed chain . . . . . 110
4.4 Mass equivalent principal open chain of three elements . . . . . . . . . . . 114
4.5 Principal vector linkages of closed chains of n elements . . . . . . . . . . 129
4.6 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Principal vector linkage architecture with similar linkages . . . . . . . . . . 135
5.1 Architecture with CoM in invariant point in a similar linkage . . . . . . 135
5.2 Conditions for similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Force balance conditions from mass equivalent principal chain . . . . . 142
5.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6 Principal vector linkages for inherent shaking moment balance . . . . . 161
6.1 Moment balance conditions of open-chain principal vector linkages 161
6.1.1 Moment balance of a 2-DoF pantograph . . . . . . . . . . . . . . . . . 162
6.1.2 Moment balance of three principal elements in series . . . . . . 165
6.1.3 Moment balance of four principal elements in series . . . . . . . 171
6.2 Moment balance conditions of closed-chain principal vector linkages178
6.3 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7 Synthesis of inherently dynamically balanced (IDB) mechanisms . . . . 183
7.1 Approach for synthesis of IDB mechanisms . . . . . . . . . . . . . . . . . . . . 183
7.2 Synthesis of an IDB 2-DoF grasper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3 Synthesis of multi-DoF IDB manipulators . . . . . . . . . . . . . . . . . . . . . . 186
7.4 Synthesis of large-size balanced devices . . . . . . . . . . . . . . . . . . . . . . . 191
7.5 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8 Experimental evaluation of a dynamically balanced redundant
planar 4-RRR parallel manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 Approach to the evaluation and comparison of a balanced
manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.3 Design of the DUAL-V manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.4 Inverse dynamic model and validation with simulation model . . . . . . 201
8.4.1 Inverse dynamic model to derive the actuator torques . . . . . . 201
8.4.2 Simulation and validation of the inverse dynamic model . . . . 206
8.5 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.6 Experiments and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.7.1 Shaking forces and shaking moments . . . . . . . . . . . . . . . . . . . 212
8.7.2 Sensitivity to balance inaccuracy and payload . . . . . . . . . . . . 216
8.7.3 Actuator torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.7.4 Bearing forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.7.5 Evaluation method and experimental setup . . . . . . . . . . . . . . . 218
8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9 Reflection on the design of inherently balanced mechanisms . . . . . . . . 221
10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
A The work of Otto Fischer and the historical development of his
method of principal vectors for mechanism and machine science . . . . 229
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.2 Otto Fischer and his works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.3 The method of principal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
A.4 Applications by Otto Fischer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
A.5 Development and application by other researchers . . . . . . . . . . . . . . . 236
A.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
About the author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
