Safe ZYZ Regrasp Strategies

• Safe ZYZ regrasp strategies are planning methods that use Z–Y–Z Euler angles to define end-effector orientation while embedding explicit safety conditions like singularity avoidance.
• They combine multi-modal motion primitives and Markov Decision Process formulations to enforce kinematic, dynamic, and collision checks during complex regrasp actions.
• These strategies ensure robust operation by integrating singularity safeguards, friction and torque constraints, and uncertainty-aware control in both simulation and real-world executions.

Safe ZYZ regrasp strategies are regrasp planning and execution methods in which object or end-effector orientation is represented by Z–Y–Z Euler angles $(\alpha,\beta,\gamma)$ and coupled to explicit safety conditions during manipulation. In the cited work, safety is enforced through singularity-avoidance guards for $\beta \rightarrow \pm 90^\circ$, kinematic-feasibility and dynamic-feasibility checks, collision avoidance, workspace containment, grasp-torque and friction constraints, and, in a related line of work, uncertainty-robust external-contact pushing for fixtureless fixturing. Across these formulations, regrasp is not treated as a single motion primitive but as a sequence involving sliding, lifting, transfer, approach, release, or pushing under manifold, contact, and reachability constraints (Hu et al., 2023, Oh et al., 29 Aug 2025, Chavan-Dafle et al., 2018).

1. Formal problem setting

A central motivation for regrasp planning is that grasping algorithms can predict a candidate of grasp poses, even for unseen objects, yet some of the predicted grasp poses to stably lift a target object may not be directly approachable due to workspace limitations. In such cases, the robot will need to re-grasp the desired object to enable successful grasping on it. The 2023 multi-modal planner formulates this as a Markov-Decision Process in which a state is a tuple

$s=(q,M,o),$

where $q\in\mathbb{R}^n$ is the full robot configuration, $M$ is the mode or manifold index, and $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$ is the object pose parametrized by position plus Z–Y–Z Euler angles. The planner distinguishes the foliations $F^{slide}$, $F^{transfer}$, and $F^{re\text{–}grasp}$, and it treats the current grasp-ID $g$ as implicit whenever $\beta \rightarrow \pm 90^\circ$0 (Hu et al., 2023).

Within that formulation, every action is a parameterized primitive

$\beta \rightarrow \pm 90^\circ$1

with $\beta \rightarrow \pm 90^\circ$2. The planner therefore places regrasp in a broader multi-modal manipulation setting rather than restricting it to a single wrist reorientation or a single grasp swap. A related but execution-oriented formulation appears in the 2025 Soma-cube system, where safe ZYZ regrasp is embedded in legal-action masked DQN and in the control layer of a Doosan M0609 collaborative robot, so that orientation decisions are filtered by reachability and safety before execution (Oh et al., 29 Aug 2025).

A broader related context is provided by fixtureless fixturing, where regrasping is achieved by pushing the object against external contact or contacts in the environment such that the external contact keeps the object stationary while the fingers slide over the object. That line of work does not center Z–Y–Z angles, but it addresses the same core problem of safe reconfiguration of grasp state under explicit contact-mechanics constraints (Chavan-Dafle et al., 2018).

2. Z–Y–Z orientation model and singularity-safe orientation handling

In the 2025 formulation, the end-effector target orientation is represented as

$\beta \rightarrow \pm 90^\circ$3

with the standard matrix forms for $\beta \rightarrow \pm 90^\circ$4 and $\beta \rightarrow \pm 90^\circ$5. This provides a direct orientation parameterization for regrasp planning and for the orientation component of the hierarchical DQN, where “ori” corresponds to one of the 116 Z–Y–Z orientations (Oh et al., 29 Aug 2025).

The principal safety issue associated with this representation is that Z–Y–Z suffers gimbal lock when $\beta \rightarrow \pm 90^\circ$6. The cited work gives two classical detection or prevention mechanisms. The first is a proximity index on $\beta \rightarrow \pm 90^\circ$7,

$\beta \rightarrow \pm 90^\circ$8

and if $\beta \rightarrow \pm 90^\circ$9, the pose is treated as near-singular. The second is Jacobian-based manipulability in the sense of Yoshikawa,

$s=(q,M,o),$0

with either a lower bound $s=(q,M,o),$1 or, equivalently, an upper bound on the condition number

$s=(q,M,o),$2

In the paper, the simpler $s=(q,M,o),$3-proximity guard is used for real-time: if $s=(q,M,o),$4 or $s=(q,M,o),$5, then $s=(q,M,o),$6 is clamped to $s=(q,M,o),$7 and regrasp logic is triggered (Oh et al., 29 Aug 2025).

The 2023 planner shows how Z–Y–Z variables can also be inserted directly into the MDP state and transition cost. It explicitly includes $s=(q,M,o),$8 in $s=(q,M,o),$9 and permits an orientation-variation penalty

$q\in\mathbb{R}^n$0

It also allows hard limits by discarding any action for which any $q\in\mathbb{R}^n$1. In $q\in\mathbb{R}^n$2 it enforces “vertical orientation,” i.e. $q\in\mathbb{R}^n$3 so the cup or object does not tilt (Hu et al., 2023).

Taken together, these formulations show that safe ZYZ regrasp is not merely a coordinate convention. The orientation representation directly enters state construction, feasibility tests, transition costs, and singularity guards. A plausible implication is that the usefulness of Z–Y–Z in regrasp planning depends less on the parametrization alone than on whether the planner and controller explicitly protect against the $q\in\mathbb{R}^n$4 regime.

3. Safety conditions in multi-modal planning

The 2023 MDP-based planner makes safety an action-feasibility property before optimization. Every primitive is feasible only if kinematic-feasibility and dynamic-feasibility hold, specifically $q\in\mathbb{R}^n$5, $q\in\mathbb{R}^n$6, and for any intermediate waypoint, friction-torque and contact constraints hold. For sliding, the object is dragged on the table plane with $q\in\mathbb{R}^n$7; for lift, the motion is a vertical pull-up with $q\in\mathbb{R}^n$8 and orientation locked; and for approach or release, the motion is between a pre-grasp pose and a stable placement. These constraints define the admissible regrasp graph before value iteration is applied (Hu et al., 2023).

Contact stability is enforced by a Coulomb-friction grasp condition. The grasp is stable if

$q\in\mathbb{R}^n$9

and no action is added to the action set unless it satisfies this at every waypoint. The same summary introduces a torque-margin penalty

$M$0

that can be added to the reward if violated, or the action can instead be marked infeasible. The planner therefore couples safety to both hard feasibility and soft cost shaping (Hu et al., 2023).

Collision avoidance and workspace limits are also explicit. Every trajectory is post-checked against the environment’s collision geometry. Table-plane containment is required for $M$1 with $M$2, whereas $M$3 requires $M$4, and hands must remain in the robot’s reachable polytope. The reward uses a negative-step cost with terminal rewards,

$M$5

so that regrasp plans are optimized not only for goal achievement but also for fewer steps and smaller orientation swings (Hu et al., 2023).

The transition model further exposes safety and uncertainty at the planning level. If $M$6 is the success probability of action $M$7, then

$M$8

with all other entries zero. Initially, $M$9 for inter-manifold actions and $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$0 for intra-manifold actions until tried. On failure, $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$1 and similar slide actions are also reduced; on success, $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$2; and actions with $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$3 are pruned, with $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$4 given as an example. This makes safety adaptive: the planner updates its belief about which regrasp actions are reliable and removes hopeless actions during replanning (Hu et al., 2023).

4. Execution-level safe regrasp on the Doosan M0609

The 2025 safe ZYZ regrasp routine is an execution algorithm for singularity-safe motion planning on a Doosan M0609 collaborative robot. Its input includes the current joint state $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$5, the desired end-effector pose $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$6, a singularity threshold $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$7 rad, a manipulability threshold $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$8, the robot model with joint limits, and a collision-checking module. The procedure first extracts $o=(x,y,z,\alpha,\beta,\gamma)\in SE(3)$9, computes $F^{slide}$0, and branches according to singularity risk (Oh et al., 29 Aug 2025).

If $F^{slide}$1 or $F^{slide}$2, the controller clamps $F^{slide}$3 and constructs an intermediate wrist-roll pose

$F^{slide}$4

It then solves IK for $F^{slide}$5 and for

$F^{slide}$6

subject to joint-limit, collision, and manipulability checks. If a valid sequence exists, it executes the two-step motion $F^{slide}$7 and returns success. If not, the fallback is a systematic 6-step regrasp sequence: apply small $F^{slide}$8-rotation on wrist J5 $F^{slide}$9, retract 50 mm in Z and open gripper, rotate tool by $F^{transfer}$0, close gripper, attempt IK to $F^{transfer}$1, and, if success and safe, execute and exit the loop. If no singularity risk is detected, the system attempts direct IK to $F^{transfer}$2 and executes a direct motion if the result is safe; otherwise it also falls back to the 6-step sequence (Oh et al., 29 Aug 2025).

This execution layer is specialized to the Doosan M0609. The cited specification includes joint limits for $F^{transfer}$3, $F^{transfer}$4, and $F^{transfer}$5, as well as velocity limits

$F^{transfer}$6

The regrasp fallback uses Cartesian-linear moves for safe retraction by moving the tool +Z by 50 mm, followed by pure wrist rotations on joint 5. Trajectory generation uses MoveIt2’s time-parameterized planner with jerk limits of 2 m/s$F^{transfer}$7 on linear segments and 10 rad/s$F^{transfer}$8 on joints, and gripper force is closed-loop controlled to 15 N target with $F^{transfer}$9, minimizing slip during mid-air rotations (Oh et al., 29 Aug 2025).

The paper also reports an illustrative example trajectory in which an orientation change from $F^{re\text{–}grasp}$0 to $F^{re\text{–}grasp}$1 is achieved via a 2-step rotation. The direct wrist-only trajectory fails IK, whereas the intermediate regrasp rotates tool Z by $F^{re\text{–}grasp}$2, lifts 50 mm in Z, recloses the gripper, and rotates to the final $F^{re\text{–}grasp}$3 while maintaining manipulability $F^{re\text{–}grasp}$4 throughout. This concrete example clarifies that “safe” in the execution sense includes both singularity avoidance and maintenance of acceptable manipulability (Oh et al., 29 Aug 2025).

5. Integration with decision-making and constrained task planning

In the 2023 multi-modal formulation, planning proceeds by value iteration:

$F^{re\text{–}grasp}$5

followed by extraction of the greedy policy $F^{re\text{–}grasp}$6 and policy rollout from $F^{re\text{–}grasp}$7 until hitting $F^{re\text{–}grasp}$8 or $F^{re\text{–}grasp}$9. The practical pipeline is explicitly enumerated: segment the object, estimate its COM, run ContactGraspNet to get grasps $g$0, label grasps high or low by $g$1 versus $g$2, construct the three foliations, enumerate inter- and intra-manifold actions, check kinematic, dynamic, and collision feasibility, build the MDP $g$3, run value iteration, extract the task sequence of manifold-constrained motion queries, call a constrained Lazy-PRM with projection onto manifold to compute each trajectory, and, on failure, update the $g$4 values, prune hopeless actions, and re-compute $g$5. The stated result is a motion plan that never violates the friction-stability condition, respects Z–Y–Z orientation limits, avoids collisions and stays in the robot’s workspace, and is optimized in expectation for few failures and minimal reorientations (Hu et al., 2023).

In the 2025 system, safe ZYZ regrasp is embedded in the legal-action mask of the assembly MDP. The paper defines

$g$6

and states that, in practice, “reachable” now includes whether a safe Z–Y–Z regrasp sequence can be found. If pure IK to $g$7 fails due to near-singularity, the controller invokes the safe regrasp procedure; if that finds a kinematically and collision-free path, the action remains in the mask, otherwise it is masked out. The safety-guard logic lives entirely inside the environment.step(...) call, so the DQN never “sees” a crash—illegal regrasp merely becomes an illegal action (Oh et al., 29 Aug 2025).

The same work places safe regrasp inside a hierarchical architecture that decomposes Q-function estimation into orientation and position components, reducing computational complexity from $g$8 to $g$9 while maintaining solution completeness. This suggests a division of labor between symbolic or discrete decision layers and continuous safety layers: the RL system chooses an orientation-position pair, while the regrasp controller certifies whether that choice is executable without singularity, collision, or joint-limit violation (Oh et al., 29 Aug 2025).

6. Robust external-contact variants, empirical results, and interpretation

A distinct but closely related safety notion appears in fixtureless fixturing. There, the robot pushes the object against external contact so that the external contact keeps the object stationary while the fingers slide over the object. The mechanics are formulated through a generalized friction cone at a pusher, a quasi-static force balance for a sticking push under gravity-balancing orientation, and a robust motion cone in twist space. The key parameter-free condition is

$\beta \rightarrow \pm 90^\circ$00

which is then mapped to the convex cone of achievable instantaneous twists

$\beta \rightarrow \pm 90^\circ$01

or equivalently to the half-space form

$\beta \rightarrow \pm 90^\circ$02

Because, in the gravity-balanced formulation, the terms $\beta \rightarrow \pm 90^\circ$03, $\beta \rightarrow \pm 90^\circ$04, and $\beta \rightarrow \pm 90^\circ$05 cancel out, the only remaining parameter is the pusher friction cone, which is conservatively approximated using $\beta \rightarrow \pm 90^\circ$06. The resulting robust push-set yields safe sticking pushes under the modeled uncertainties (Chavan-Dafle et al., 2018).

Planning in that framework grows a T-RRT* tree in the 3D grasp-pose space $\beta \rightarrow \pm 90^\circ$07, and the key safety check is projectOntoCone, which ensures that the small twist applied lies inside a robust motion cone $\beta \rightarrow \pm 90^\circ$08, thus guaranteeing a sticking push under all modeled uncertainties. The cost function

$\beta \rightarrow \pm 90^\circ$09

biases the planner against pusher switch-overs, because each switch incurs noise. Although this framework is not expressed as a Z–Y–Z wrist singularity problem, it is directly relevant to safe regrasp strategies because it formalizes regrasp safety through uncertainty-invariant contact mechanics rather than through Euler-angle guards (Chavan-Dafle et al., 2018).

The empirical results across the cited works illustrate several distinct notions of success. In fixtureless fixturing, planner time was 0.39 s for a 20 mm horizontal offset regrasp, 0.65 s for a general $\beta \rightarrow \pm 90^\circ$10 regrasp on a rectangular prism, and 0.60 s for a T-shaped nonconvex object; the T-shaped object showed 20 consecutive open-loop trials under varied grasp forces with zero failures, and changing $\beta \rightarrow \pm 90^\circ$11 via grasp-force variation had no effect on outcome (Chavan-Dafle et al., 2018). In the MDP-based multi-modal planner, the authors report improved performance in both simulation and the real world for pick-and-place tasks (Hu et al., 2023). In the Doosan M0609 system, curriculum learning across three progressive difficulty levels achieved 100\% success rate for Level 1 within 500 episodes, 92.9\% for Level 2, and 39.9\% for Level 3 over 105,300 total training episodes, and embedding the safe-regrasp routine in the action-masking and execution layers raised motion-success from 54 % to 96 % in real-world trials while enabling placement of every Soma piece in all 116 orientations without gimbal-lock failures (Oh et al., 29 Aug 2025).

The cited work therefore indicates that safe ZYZ regrasp should not be reduced to a singularity workaround alone. In the MDP formulation, safety includes friction-stability, collision avoidance, workspace containment, and expected-failure minimization. In the Doosan execution stack, safety includes $\beta \rightarrow \pm 90^\circ$12-proximity guards, manipulability thresholds, joint limits, collision checks, smooth trajectory generation, and slip-aware gripper control. In fixtureless fixturing, safety is expressed as robustness of sticking pushes under uncertainty. A plausible implication is that “safe ZYZ regrasp” is best understood as one member of a broader family of safe regrasp strategies in which orientation representation, contact mechanics, and task-level action filtering are tightly coupled rather than isolated design choices.