We summarize the state of the art for base placement optimization and discuss related and established benchmarks. Optimizing the base pose can be considered part of laying out a manufacturing system. Moslemipour et al. (2012) discussed different methods, mostly from derivative-free black-box optimization.

Noticeable exceptions are Son and Kwon (2019) and Baumgärtner et al. (2024), which formulate base placement optimization as (convex) optimization problems. Son and Kwon (2019) exploited the fact that the inverse kinematics of industrial robots with spherical wrists can be decomposed into a positioning and orientation sub-problem. Baumgärtner et al. (2024) optimized the entire robot structure, but particularly the robot base, jointly in a trajectory tracking problem. They modeled the base as a set of zero-velocity joints and solved the whole optimization via collocation.

We group the remaining methods into exhaustive search, capability maps, genetic algorithms, and Bayesian optimization.

ES: This method optimizes the base position of an industrial robot by evaluating all base positions on an equidistant grid (Lechler et al., 2021).

We present the first work comparing base placement optimization algorithms on a set of benchmarks and adapt Adam (Kingma and Ba, 2015), an SGD method, to the base placement optimization problem. In particular, we

compare base placement optimization methods [GA, BO, random sampling (RS), and SGD] with respect to cost convergence and success rate;

In Section 2, we define the optimization problem and introduce the evaluated methods. Section 3 states the setup for our evaluation and summarizes the results. These are discussed in Section 4, which is followed by a conclusion in Section 5.

As a baseline, we implement a dummy base placement optimization that keeps the nominal pose of the robotic task. Additionally, we consider an RS optimizer that uniformly samples poses from the allowed set B . With an increasing number of sampled points, it approximates the ES method surveyed in Section 1.1.

GAs belong to the set of black-box optimization algorithms. Within a GA, a population of individual solution candidates encoded as a list of genes is optimized. Those genes are our parameter vectors, defined in Equations 4, 5, and are altered by

mutation, which locally alters a single gene or several genes of an individual;

The mutation operator uniformly selects single values b i , i ∈ [ 1 , … , a ] to alter and adds a random number from a scalar normal distribution with zero mean and unit standard deviation N ( 0,1 ) to them. Cross-over selects the first n ∈ [ 1 , … , a ] entries in one gene vector and combines it with the last a − n elements from another individual. The selection is based on the negative cost function − J C ( B ) , which is considered the fitness of each individual.

BO is another common method within derivative-free or black-box optimization algorithms. Similar to Kim et al. (2021), our BO uses a Gaussian process J ̂ C ( B ) to approximate the cost of i previously tested base poses { J C ( B 1 ) , … , J C ( B i ) } . The next base pose B i + 1 to explore is sampled from J ̂ C ( B ) based on the exploration/exploitation hyperparameter ξ . The resulting cost J C ( B i + 1 ) can, in turn, be used to refine J ̂ C ( B ) . For base placement optimization, we define J ̂ C ( B ) using B( b ̄ ) ∈ R a following Equations 4, 5 and use it to fit the scalar cost value J C ( B ) . At the start, BO commonly samples n i n i t ∈ N _>0 points, e.g., randomly or from low-discrepancy sequences, such as the Hammersley set (Wong et al., 1997). The initial sampling method is also determined via hyperparameter tuning.

SGD is a gradient-based method that gained prominence for training deep neural networks. Here, it serves as the outer method of a two-level optimization. A current state-of-the-art SGD algorithm is Adam (Kingma and Ba, 2015). Given a manipulator with known forward kinematics FK : R n DoF → S E ( 3 ) , i.e., a function returning the end-effector pose of a robot given its joint angles q ̄ ∈ R n DoF and a distance function δ : S E ( 3 ) × S E ( 3 ) → R ≥ 0 , we can use numerical inverse kinematics (IK) to find configurations q ̄ i close to desired goal poses g i ∈ S E ( 3 ) (Siciliano and Khatib, 2016; Section 2.7) (inner method). We use Adam to optimize the base pose, such that δ is reduced for the found configurations q ̄ i . Starting at a random base pose determined by b ̄ ∈ R a , we run the following two-level optimization loop n A d a m ∈ N _>0 times.

We move the robot to B ( b ̄ ) and find the closest IK solutions q ̄ i ∈ R n DoF to all goals g i ∈ G according to a distance function (Equation 6)

within a maximum of n I K ∈ N _>0 steps. We terminate if all IK solutions are within the tolerance of each goal.

2. We calculate the gradient of δ with respect to the base parameters b ̄ at each found IK solution q ̄ i (Equation 7):

The algorithm is restarted from other random initial guesses while there is time left to escape local minima.

All base placement optimization methods have access to the same task solver that judges whether a suggested base pose B allows the robot R to fulfill all goals G of a given task. This solver first applies step-by-step elimination (Althoff et al., 2019, p. 7–8) to reject base poses further away from any goal than the maximum length of the robot or those without (collision-free) inverse kinematic solutions. If these filters pass, we run RRT-Connect (Kuffner and La Valle, 2000) and the trajectory generator proposed by Kunz and Stilman (2012) to find a collision-free and feasible trajectory x ( t ) for the robot positioned at B . The overall process is summarized in Figure 2. We calculate the cost function J C based on the found trajectory. If any step fails, a fixed failure cost J f a i l ∈ R is returned to the optimizer.

The overall runtime of the task solver varies depending on which filter rejects the suggested base pose. For reference, a failure in the first two steps can be determined within milliseconds as these only need a few forward kinematic calculations. Collision-free inverse kinematic solutions are identified in approximately 0.1 s, while the path planner succeeds in the order of seconds and only fails after a fixed time-out is reached.

In all our experiments, we use Timor Python (Külz et al., 2023) for kinematic calculations and collision detection. Our benchmark tasks are implemented as CoBRA tasks (Mayer et al., 2024). We created our simple set of 100 tasks by sampling three cubic obstacles on a grid around the origin and three desired poses outside those cubes that the end effector should stop at, as proposed by Whitman et al. (2020). Moreover, we tested the tuned methods on the following sets:

hard: 100 synthetic tasks using the same sampling as the simple set of tasks, with five goals and obstacles instead of three.

The tasks ⁴ are tagged with BPO24 and simple/hard/real/edge_case_hard in CoBRA.

All tasks have the following constraints:

Joints limited in position q and torque τ ,

In all cases, the goal was to minimize the cycle time J T = t f required to fulfill the given task with a given robot ⁶ . This objective has been shown to work for deployments in real-world tasks and is positively correlated with other costs, such as energy consumption (Mayer and Althoff, 2025). If the solver cannot find a solution trajectory for a suggested base pose, a default failure cost of J f a i l = 20 s is returned, which is well above the possible cycle time. All experiments were run on a 64-core AMD EPYC 7742 clocked at 2.25 GHz. The inverse kinematic solver and Equation 6 in SGD used a weighted sum of Euclidean distance and rotation angle between poses δ , with the default weight 5  mm / 1.8 ° of Timor Python (Külz et al., 2023). We, therefore, keep this weighting parameter fixed.

The SGD optimizer used Adam and automatic differentiation implemented using PyTorch 2.2.1 (Paszke et al., 2019). The GA was implemented with PyGAD (Gad, 2023). BO used the implementation provided by scikit-optimize ⁷ . SGD, GA, and BO require setting hyperparameters to tune them for specific applications, in contrast to dummy and RS. We used Optuna (Akiba et al., 2019) in its default setting with the tree-structured Parzen estimator (TPE) optimizer (Bergstra et al., 2011) and median pruner to minimize the mean cost on the first 70% of simple tasks. The base placement optimization was limited to finding the best position for the robot base by using Equation 4. In total, we ran 400 trials for each of the three tunable algorithms with Optuna. The optimized hyperparameters are listed in the Supplementary Material. For RS, we did not identify any tunable parameter; it always uses uniform distribution sampling from the whole domain of b ̄ .

The final comparison of tuned algorithms was performed on a distinct set of tasks not used during hyperparameter optimization, i.e., the last 30% of the simple set and the complete sets hard, real, and edge. For all sets in addition to simple, we had to increase the failure penalty to J f a i l = 50 s as the additional goals and more complex obstacles need, on average, longer cycle times (see average cycle times in Figure 3). The comparison with tasks outside the simple set is particularly important as it shows the generalization potential of each algorithm. Algorithms are compared in a fixed-budget setting: they obtain t C P U = 1200 s = 20 min ⁸ per task to explore possible base poses. This time limit encompasses all optimization steps, including the internal calculations for each optimization method, running the provided filters, and, upon success, completing the path planning. The final score is the cost of the best solution, and we present statistics over five distinct seeds.

First, we present the convergence of the minimum cost found (top) and the success rate (bottom) on the hard (Figure 3a) and a real set of tasks (Figure 3b). The success rate is the fraction of tasks and seeds for which each base placement optimization algorithm can find a valid base pose within the optimization time as the x-axis. The first column of each subfigure lists the results obtained by optimizing only the base position (see Equation 4) used during hyperparameter optimization; the second column lists the results when optimizing the base position and orientation (see Equation 5). The mean and 95% confidence interval calculated via bootstrapping over all tasks and seeds are shown for each method.

Second, we summarize the final average best cycle time and the success rate of each method across all sets of tasks in Table 1. To judge significance, we provide the mean expected cost/success rate over all found solutions, along with 95% confidence intervals calculated using bootstrapping. Therefore, the numbers provided in Table 1 for the hard task set show the mean and confidence interval at t C P U from Figure 3. We highlight mean values outside the confidence interval of all other methods in the same row as worse or better. Furthermore, we conduct a Wilcoxon signed-rank test with Bonferroni p-value corrections over all tasks used in the evaluation to identify significant differences between algorithms; the full test statistics are provided in Supplementary Tables S4, 5. The hard, real, and edge sets of tasks test generalization as they have different distributions of goal poses and obstacles compared to the simple set of tasks used for hyperparameter optimization. We omit the dummy optimizer as it can only solve the simple set of tasks in 47% of cases and the hard set of tasks in 18% while failing to solve any task from the other sets.