Beyond Deviation Magnitude: Phase-Joint Directional Characterization of Physics-Validated Manipulation Trajectories from a Single Demonstration

Imitation learning enables robots to acquire complex manipulation behaviors from human or expert demonstrations. However, collecting a sufficiently large set of demonstrations on physical robots is time-consuming and costly. The environment must be reset, an operator must repeatedly intervene, and failed trials often require recollection. When object locations or task conditions change, the same collection process must be repeated. Consequently, methods that obtain more diverse training data from a small number of real demonstrations are needed. To address this limitation, prior work has generated new data by transforming or reconstructing existing demonstrations. MimicGen decomposes demonstrations into task segments and reassembles them for new object configurations [5], while DemoGen expands a single demonstration across diverse scene conditions [7]. Action-view augmentation and SART have likewise proposed methods for producing more varied execution trajectories from limited demonstrations [3], [4]. As these approaches make it possible to create more execution experience from sparse data, evaluating how the generated trajectories differ from one another has become increasingly important. Trajectory diversity is commonly evaluated using a single global quantity, such as an inter-trajectory distance or variance. Mitrano and Berenson treated diversity, together with validity and relevance, as an important criterion in manipulation data augmentation. Such metrics are useful for comparing the overall spread of datasets; however, a single scalar value does not reveal at which task phase the differences arise or which joints contribute to them. Manipulation trajectories involve multiple joints whose roles vary over time. Even within a pick-and-place task, approach, lift, transport, and place serve different objectives, and a deviation of the same magnitude can have different implications depending on the phase and joint in which it occurs. Moreover, two trajectories that deviate from a seed trajectory in opposite directions may be assigned the same unsigned magnitude. Overall deviation magnitude alone may therefore be insufficient to characterize differences among successful trajectories. This study analyzes variation among successful manipulation trajectories from the perspectives of task phase, joint, and direction. For trajectories generated from a single demonstration and retained after physics validation, we compare both the magnitude and the direction of deviation from the seed and examine their associations with task outcome. The analysis asks whether variation among successful trajectories can be explained by deviation magnitude alone, or whether it is necessary to preserve where, in which joint, and in which direction the variation occurs.

Let q_seed(t) denote a seed demonstration and q_i(t) the i-th successful trajectory that passes physics validation. The objective is not only to construct a set of successful trajectories, but also to decompose the variation observed around the seed according to task phase p and joint j and to analyze how this variation is associated with the final object placement error. The problem is formulated at two levels. The first is structured trajectory generation, in which allowable perturbations are selectively allocated using task phase and joint role. The second is successful-variation characterization, in which magnitude-only and direction-preserved deviations are computed for physics-validated trajectories and their associations with terminal accuracy are compared.

For each task phase p, the cumulative motion of joint j, denoted L_p,j, is defined as

L_p,j = Σ_k |q_k,j − q_k−1,j| (1)

To reduce the influence of phase duration, the cumulative motion is normalized by the phase duration T_p to obtain the joint motion rate

R_p,j = L_p,j / T_p (2)

To distinguish repeated back-and-forth motion within a phase from motion that proceeds consistently in a particular direction, the directional efficiency η_p,j is defined as

η_p,j = |q_e,j − q_s,j| / (L_p,j + ε) (3)

The motion rate is then normalized to compare the relative contribution of each joint within the same phase. The phase-joint importance is computed as the product of the normalized motion rate and directional efficiency

I_p,j = R̃_p,j · η_p,j (4)

Based on this measure, the relative importance of each joint is determined and combined with task semantics to classify augmentation freedom as LOCK, LOW, MEDIUM, or HIGH. During CLOSE, GRASP_HOLD, and RELEASE, where contact between the object and the end effector is directly established or terminated, all joints are set to LOCK. APPROACH and TRANSPORT are assigned relatively larger freedom, whereas LIFT and PLACE are assigned more restricted freedom.

To analyze variation within the successful trajectory set, the root-mean-square deviation for trajectory i, phase p, and joint j is first defined as a magnitude-only metric

M_i,p,j = √[(1 / N_p) Σ_k∈p (q_i,k,j − q_seed,k,j)²] (5)

Here, N_p denotes the number of time steps contained in phase p. The RMS deviation represents the average magnitude by which a joint trajectory departs from the seed during that phase. Because squaring removes the sign of the deviation, however, deviations in the positive and negative directions relative to the seed are represented by the same magnitude. To preserve directional information, the mean signed deviation is defined as the direction-preserved metric

S_i,p,j = (1 / N_p) Σ_k∈p (q_i,k,j − q_seed,k,j) (6)

If S_i,p,j is less than zero, the joint lies, on average, in the negative direction relative to the seed during the phase; if S_i,p,j is greater than zero, it is biased in the positive direction. Thus, M_i,p,j measures how far the trajectory departs from the seed, whereas S_i,p,j preserves the direction of that departure. Applying both metrics to the same successful trajectories and phase-joint pairs allows us to determine whether the association with terminal accuracy is captured by deviation magnitude alone or becomes clearer when directional information is retained.

To visualize temporal variation among successful trajectories, all trajectories are aligned on normalized trajectory time, and Q₀₅, Q₅₀, and Q₉₅ are computed at each time-joint location. The interval from Q₀₅ to Q₉₅ is used as the Successful Trajectory Envelope, or STE

STE_j(t) = [Q₀₅(q_j(t)), Q₉₅(q_j(t))] (7)

The STE does not represent the true physical feasibility boundary of the robot. Rather, it is defined here as the empirical range of variation formed by trajectories that were successfully executed under the specific augmentation procedure and physics-validation conditions used in this study.

The proposed system follows the sequence Single Demonstration, Task Phase Segmentation, Phase by Joint Analysis, Selective Trajectory Augmentation, Candidate Trajectories, Physics Validation, Successful Trajectory Set, and Successful Variation Analysis. The central idea is to construct a successful set through generation and validation and then analyze its variation by separating magnitude from direction.

Experiments used an SO-101 robot consisting of five arm joints and a gripper. Joint states for the pick-and-place demonstration were recorded at approximately 50 Hz, and the refined seed demonstration contained 4,787 frames. The full trajectory was segmented into preparation, approach, gripper close, grasp hold, lift, transport, place, release, and termination. Among these phases, the motion characteristics of APPROACH, LIFT, TRANSPORT, and PLACE were used to determine augmentation freedom.

The maximum perturbation bounds for LOW, MEDIUM, and HIGH freedom were set to 1, 3, and 6 degrees, respectively. The actual phase-joint perturbation amplitude A_p,j was determined as the smaller of the net joint displacement D_p,j observed in the seed demonstration and the freedom-level bound B_c

A_p,j = min(D_p,j, B_c(p,j)) (8)

For the joint order [Shoulder Pan, Shoulder Lift, Elbow Flex, Wrist Flex, Wrist Roll], A_p,j for APPROACH, LIFT, TRANSPORT, and PLACE was set to [2.50, 6.00, 6.00, 1.00, 6.00] degrees, [0.03, 2.96, 0.06, 1.71, 0.05] degrees, [6.00, 3.00, 6.00, 1.00, 6.00] degrees, and [0.05, 1.43, 0.03, 0.84, 0.02] degrees, respectively. During CLOSE, GRASP_HOLD, and RELEASE, perturbation of every joint was restricted to 0 degrees.

A C²-continuous envelope was applied so that perturbations would not introduce discontinuous changes in position, velocity, or acceleration at phase boundaries

E(s) = 64s³(1 − s)³, 0 ≤ s ≤ 1 (9)

The final candidate trajectory is generated as

q_i,k,j = q_seed,k,j + A_p,jE(s_k)r_i,p,j,k (10)

where r is a low-frequency perturbation component normalized to the interval [−1, 1]. This formulation makes the perturbation smoothly converge to zero at each phase boundary while producing distinct phase-joint variation structures.

The generated candidate trajectories were executed in an SO-101 physics environment implemented in Isaac Sim. During each execution, object position was tracked and success was determined using lift, transport, placement, and release conditions. The object size was 30.9 mm, and only trajectories satisfying all constraints on lift height, minimum transport height, placement height, and release height were retained as successful trajectories. The final planar distance to the target was not included in the binary success criterion; instead, it was used as a continuous placement-accuracy metric

e_i = ‖x_i^final − x_target‖₂ (11)

The preceding comparative evaluation considered Uniform, Phase-Gated, Phase-Aware Joint-Uniform, and the Proposed method. For each method, 200 trajectories that passed physics validation were collected, after which 150 were randomly selected; this sampling procedure was repeated 100 times. The successful-variation analysis in this paper uses 150 trajectories from the physics-validated successful set to examine the relationship between phase-joint deviation and final target error.

Associations between final target error and both magnitude-only RMS deviation and mean signed deviation were evaluated using Spearman rank correlation. To mitigate the multiple-comparison problem arising from exploration of numerous phase-joint and metric combinations, Benjamini-Hochberg false discovery rate correction was applied. For the strongest association, robustness was examined using a bootstrap confidence interval and outlier trimming. Directional asymmetry was assessed by separating trajectories into S<0 and S≥0 groups according to the sign of the signed deviation and comparing their final target-error distributions. Because this analysis evaluates observational associations, a perturbation in a particular direction is not interpreted as a causal source of improved accuracy.

Phase-joint-selective augmentation was applied to a single SO-101 pick-and-place demonstration, and the resulting candidate trajectories were executed in Isaac Sim. Only trajectories satisfying the lift, transport, placement, and release conditions were retained in the successful set. In the preceding comparative experiments, the proposed method maintained a level of trajectory diversity comparable to uniform-type baselines while achieving a lower final target error, and joint-wise variation remained differentiated within the successful set. The analysis below therefore concerns differences among trajectories that satisfied the same task objective in the physics environment, rather than differences among numerically generated trajectories alone. When the successful trajectories were aligned on a normalized time axis, a bounded variation envelope formed around the seed, with a width that varied across time and joints. This indicates that the successful trajectories do not constitute an arbitrarily scattered set, but instead exhibit structured variation around the seed. Based on this observation, we examine whether simple variation magnitude and directional information differ in their ability to explain terminal accuracy.

We first tested whether larger departures from the seed were associated with larger final placement errors by computing the RMS deviation for APPROACH by Elbow Flex. In Fig. 2(a), the Spearman correlation between magnitude-only RMS deviation and final target error was ρ=−0.133 with an FDR-adjusted p value of 0.640, indicating only a weak association. Within the successful set, terminal accuracy was therefore difficult to explain using deviation magnitude from the seed alone. The result changed when the mean signed deviation was used for the same trajectories and the same phase-joint pair. In Fig. 2(b), the Spearman correlation was ρ=−0.482 and remained significant after FDR correction; the bootstrap 95% confidence interval was [−0.595, −0.352] and did not include zero. The key observation is that outcome-related structure became clearer when the direction of bias relative to the seed was preserved, rather than when only the distance from the seed was considered.

To determine whether the directional effect resulted from arbitrarily selecting a single joint, associations between mean signed deviation and final target error were computed for every combination of APPROACH, LIFT, TRANSPORT, and PLACE with the five arm joints. As shown in Fig. 3, associations that remained significant after FDR correction were concentrated in Shoulder Pan and Elbow Flex during APPROACH, whereas most other phase-joint combinations did not exhibit a clear association.

This study performed trajectory augmentation from a single real-robot demonstration while accounting for phase and joint motion characteristics, and constructed a successful trajectory set through physics validation. Rather than evaluating successful trajectories only in terms of data quantity or global diversity, we analyzed variation from the seed demonstration in terms of both magnitude and direction. The results showed that magnitude-only RMS deviation did not exhibit a clear relationship with terminal placement error, whereas preserving deviation direction revealed significant and robust associations for specific phase-joint combinations. Directional structure was particularly pronounced for Elbow Flex and Shoulder Pan during APPROACH, and these relationships remained consistent in analyses of directional asymmetry and outlier robustness. Accordingly, diversity among successful manipulation trajectories is difficult to characterize adequately in terms of variation magnitude alone. The results indicate that considering where the variation occurs, which joint exhibits it, and in which direction it develops can reveal task-accuracy-related structure more clearly. Future work should evaluate the reproducibility of this directional structure across diverse tasks, object poses, and robot embodiments and investigate its use as a criterion for data selection or adaptive augmentation. This research was supported by the Gyeongbuk ANCHOR system through the Gyeongbuk ANCHOR CENTER, funded by the Ministry of Education (MOE) and the Gyeongsangbuk-do, Republic of Korea. (2026-ANCHOR-15-107)