Location Optimization of Manipulator to Minimize Energy Considering the Path Direction

Kaho Hibino, Mitsuru Endo, Zexin Shan, Yukio Tsutsui

The 2025 IEEE/SICE International Symposium on System Integrations SII2025

ThuP1T2.7

Background

In our previous work, we applied Multidisciplinary Design Optimization (MDO) to optimize a manipulator’s link lengths, installation position, motor selection, and reduction ratio. During installation position optimization, the objective was to maximize manipulability [1]. However, conventional manipulability does not account for the end-effector’s motion direction. To address this, we proposed the Directional Manipulability Index (DMI) [2], incorporating trajectory direction into the manipulability evaluation. The results confirmed that DMI-based optimization led to an installation position better aligned with the trajectory direction than conventional manipulability.

However, manipulability alone does not consider energy consumption. In this study, we propose a new energy-based metric that accounts for trajectory direction and evaluate its effectiveness as a criterion for installation position optimization.

Method

1　DMI: Directional Manipulability Index

In our previous work, we proposed the Directional Manipulability Index (DMI) $\hat{w}_m$, a metric that extracts the trajectory-directional component of the manipulability ellipsoid, derived from the Jacobian matrix. To achieve this, we introduced a trajectory parameter $s$ along the path to extract the relevant directional component. $\hat{w}_m = \sqrt{ {\boldsymbol{J}_r}^+ \boldsymbol{J}\boldsymbol{J}^T { {\boldsymbol{J}_r}^+}^T }$ Figure 1 illustrates the relationship between the task space, joint space, and trajectory parameter space. The Jacobian matrix $\boldsymbol{J}$ serves as the transformation between the joint space and task space. Meanwhile, JrJr is obtained by differentiating the trajectory vector $r(s) \in \mathbb{R}^m$ with respect to $s$, excluding time-dependent elements. Thus, $\boldsymbol{J}_r$ can be regarded as the Jacobian matrix that maps between the task space and trajectory parameter space.

Fig. 1 Transformation Relationships Among Operational Space, Joint Space, and Parametric Variable Space

Thus, DMI represents manipulability within the trajectory parameter space and was proposed as a manipulability metric that considers trajectory direction. However, DMI only accounts for manipulability and does not incorporate energy consumption, which has been identified as a potential drawback, leading to suboptimal energy efficiency.

To address this issue, in this study, we propose a new energy-based metric that, similar to DMI, introduces a trajectory parameter to evaluate energy consumption along the trajectory direction.

2　DEI: Directional Energy Index

Similar to DMI, we consider transforming the kinetic energy of the manipulator into the trajectory parameter space using a parameterized representation.

The energy consumption generated by the motion of an ( n )-link manipulator is given by: $E_l = \sum_{i=1}^{n} \frac{1}{2} \dot{\boldsymbol{p}}_{gi}^T \boldsymbol{M}_i \dot{\boldsymbol{p}}_{gi}$ Using the transformation matrices between the task space and joint space: $\dot{\boldsymbol{p}}_{gi} = \boldsymbol{J}_i \dot{\boldsymbol{q}}$ and between the joint space and trajectory parameter space:

$\dot{\boldsymbol{q}_e}(s(t)) = \boldsymbol{J}_{jr} \dot{s}$ we express the energy consumption in the trajectory parameter space as: $E = \frac{1}{2} {\dot{s}}^T \sum_{i=1}^{e} \left\{ {\boldsymbol{J}_{jr}}^T {\boldsymbol{J}_i}^T \boldsymbol{M}_i \boldsymbol{J}_i \boldsymbol{J}_{jr} \right\} \dot{s}$ Here, assuming ( \dot{s} = 1 ) as a constant, we define ( \hat{w}_e ) as the Directional Energy Index (DEI):

$\hat{w}_e = \sum_{i=1}^{e} \left\{ {\boldsymbol{J}_{jr}}^T {\boldsymbol{J}_i}^T \boldsymbol{M}_i \boldsymbol{J}_i \boldsymbol{J}_{jr} \right\}$ DEI represents the energy displacement within the trajectory parameter space, enabling energy evaluation without requiring complex dynamic or electromagnetic calculations. Instead, it allows energy estimation directly from the Jacobian matrix, making it a metric that evaluates energy consumption purely based on trajectory information.

By optimizing the installation position to minimize DEI, we determine the optimal manipulator placement that minimizes energy consumption.

Result

In this chapter, we focus on optimizing the installation position of a manipulator using the proposed metric. The optimization targets are a two-link manipulator and a six-link manipulator, with their installation positions defined as design variables in a 2D plane and a 3D space, respectively.

The minimization problem for installation position optimization is formulated as follows: $\begin{array}{ll} \mathrm{minimize}_{\boldsymbol{x}} & N\sum^N_{i=1} f(w_i)\\ \mathrm{subject~to} & \boldsymbol{x} = (x, y, z),\\ &x \in [x_\mathrm{min}, x_\mathrm{max}],\\& y\in[y_\mathrm{min},y_\mathrm{max}], \\ &z\in[z_\mathrm{min},z_\mathrm{max}] \end{array}$ $\boldsymbol{x}$ represents the absolute coordinates of the manipulator’s base position in the global coordinate system, and $N\in\mathbb{Z}$ denotes the number of data points along the target trajectory. The function $f(w_i)$ is defined based on the selected metric.

When DEI or energy consumption is used as the metric, $f(w_i)$ corresponds to the metric itself.
When DMI is used as the metric, $f(w_i)$ is defined as the reciprocal of the metric.

By optimizing the installation position xx using each metric, we obtain the configurations that achieve minimum DEI, minimum energy consumption, and maximum DMI.

Table 1. Installation Position Optimization for a Two-Link Manipulator

	DEI	Energy consumption	DMI
Posture and position of manipulator
DEI	11.908	12.18	19.982
Energy [Nm]	1.272	1.267	2.339

Table 1 presents the optimization results for the installation position of a two-link manipulator following a linear trajectory. The left figure shows the results for DEI minimization, the center figure corresponds to energy consumption minimization, and the right figure represents the results for DMI maximization.

In the case of DMI-based optimization, which does not account for energy consumption, the major axis of the manipulability ellipsoid aligns well with the trajectory direction, leading to an installation position that enhances directional manipulability. However, this results in higher energy consumption, as observed from the computed values.

On the other hand, when DEI and energy consumption were used as optimization criteria, the results confirmed that the installation position was adjusted to reduce energy consumption, as intended. Furthermore, despite DEI-based optimization not requiring dynamic calculations, the obtained installation position resulted in almost the same energy consumption as that obtained through direct energy-based optimization, which does require dynamic calculations.

This demonstrates that DEI enables energy evaluation without requiring complex dynamic calculations, making it a computationally efficient metric for energy-aware manipulator optimization.

Conclusion

In this study, we proposed DEI (Directional Energy Index) as an energy-based metric that considers trajectory direction. We applied DEI to optimize the installation position of a two-link manipulator and confirmed that, compared to the conventional DMI, DEI leads to an installation position that reduces energy consumption.

Furthermore, compared to conventional energy minimization approaches that require full dynamic calculations, DEI achieved similar energy reduction using only kinematic calculations. This demonstrates that DEI is effective in reducing computational costs in optimization.

Moving forward, we plan to extend the Multidisciplinary Design Optimization (MDO) framework to incorporate DEI-based optimization beyond installation position, considering additional design variables for comprehensive manipulator optimization.

This work was supported by JSPS KAKENHI Grant Number 23K03755.

参考文献 - References

[1] 吉川恒夫，“ロボットアームの可操作度”, RSJ誌, vol.2, No.1, pp.63–67, 1984.

[2] 日比野圭歩, 遠藤央, Shan Zexin, 筒井幸雄, “可操作性楕円体の異方性を考慮した指標に基づく2リンクマニピュレータを対象とした軌道最適化”, RSJ2024, 1I4-04, 2024.

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