Unified topology and joint types optimization of general planar linkage mechanisms

Abstract

We propose a gradient-based topology optimization method to synthesize a planar linkage mechanism consisting of links and revolute/prismatic joints, which converts an input motion to a desired output motion at its end effector. Earlier gradient-based topology optimization methods were mainly applicable to the synthesis of linkage mechanisms connected by revolute joints only. The proposed method simultaneously determines not only the topology of planar linkage mechanisms but also the required revolute and/or prismatic joint types. For the synthesis, the design domain is discretized into rectangular rigid blocks that are connected to each other by the newly proposed revolute and prismatic joint elements, the joint states of which vary depending on the corresponding design variables. The new concept of joint elements is materialized thorough an elaborately configured set of zero-length springs whose stiffnesses vary as the functions of the design variables. Therefore, any connectivity state among unconnected, rigidly connected, revolute joint, and prismatic joint states can be represented by properly adjusting the stiffnesses. After presenting our modeling, formulation, and sensitivity analysis, the developed method is tested with verification examples. Then the developed method is extended to be able include additional shape design variables and applied to solve a realistic problem of synthesizing a finger rehabilitation robotic device. We expect the developed method to play a critical role in synthesizing a wide class of general linkage mechanisms.

Appendix: Identification of linkage mechanisms from the converged JBM systems

The process to identify mechanisms from the converged JBM systems that are explicitly composed of links and joints is illustrated in Fig. 17 for the verification problems and in Fig. 18 for the finger rehabilitation synthesis problem (this process may be automated but it is done manually for the problems solved here). The overall process can be summarized as

• STEP 1: Build post-processed JBM systems for the converged JBMs by Formulation A or B with ${\xi}_i^k\in \left[{\xi}_{\mathrm{min}}^k,\kern0.5em 0.5\right)$ and ${\xi}_i^k\in \left[0.5,\kern0.5em 1\right]$ replaced by 0 and 1, respectively.

• STEP 2: Check the inter-block connectivity using the results in Fig. 3. Especially identify the so-called mega blocks consisting of more than one block and also floating blocks that do not contribute to motion generation. This step can be repeated if necessary.

• STEP 3: Convert the processed JBM by STEP 2 to a mechanism solely consisting of links and joints. In this step, it is ensured that a link should be connected to the end effectors.

Fig. 17

The process to identify mechanisms that are explicitly composed of links and joints for the verification problems considered in Section 4.1. For the problem synthesizing a RRPP four-bar mechanism in Fig. 8a, b RRRR four-bar mechanism in Fig. 8b, and c RRPR four-bar mechanism in Fig. 8c

Fig. 18

The process to identify explicit 9-bar linkage mechanism for the finger rehabilitation device synthesis problem

For instance, let us examine the identification process illustrated in Fig. 17c. The JBM system on the left is the post-processed JBM by STEP 1. By STEP 2, Block 2 is declared as a floating block because it is connected only to Block 3 by a prismatic joint. Also, we can identify a mega block consisting of Block 7, 8, and 9. This mega block will be denoted by M-Block 7–8-9. Likewise, M-Block 4–5-6 can be identified. Then M-Blocks 4–5-6 and 7–8-9 can be merged into larger mega block, M-Block 4–5–6-7-8-9. If STEP 2 is completed, the JBM shown on the right of Fig. 17c can be obtained. Then, one can easily identify three links connected by 3 revolute joints and one prismatic joint. Note link 2 is connected to the end effector denoted by a blue circle inside M-Block 4–5–6-7-8-9. By similar procedures, the linkage mechanisms in Fig. 17a and b are identified. The process to identify the 9-link rehabilitation mechanism shown in Fig. 15 is illustrated Fig. 18. The JBM on the top of Fig. 18 is the result of STEP 1. In STEP 2, M-Blocks 8–9, 11–12, 2–3, and 5–6 are identified using the results in Fig. 3 and also two floating blocks 1 and 7 are identified. Because there are thick lines connecting end effector 2 to M-Block 8–9 and Block 14, M-Block 8–9 and Block 14 are extended to end effector 2. In identifying the final mechanism solely consisting of links and joints, the prismatic joints are properly repositioned. Their locations do not affect paths of end effectors as long as they connect the same blocks identified in STEP 1. Note that arrows indicate the initial motion directions due to the prismatic joints next to them. The third figure from the top suggests that M-Block 8–9 can be replaced by the ternary link in the bottom figure.