JAD Trajectory in Robotics & Communications

• JAD Trajectory is a hybrid state representation that couples joint angles with distance geometry to model kinematics and beamforming behaviors.
• The concept is context-dependent, varying across robotics, near-field localization, and communication through distinct independent variables and interpretations.
• Methodologies involve techniques like EDM completion, Riemannian optimization, and joint-path parameterization to integrate angle and distance metrics.

Searching arXiv for recent and directly relevant papers on Joint Angle-Distance / related trajectory formulations. “Joint Angle-Distance (JAD) trajectory” is not a uniformly standardized term in the cited arXiv literature. The available uses suggest two principal interpretations. In robotics, pose estimation, and kinematic modeling, a “JAD trajectory” (Editor’s term) is best understood as a trajectory whose state couples joint-angle coordinates with distance-based geometric quantities such as prismatic displacements, limb lengths, or Euclidean distance matrix (EDM) entries (Marić et al., 2020, Fried et al., 2024, Bilić et al., 2023). In near-field localization and frequency-diverse beamforming, the term is used more literally for a locus in joint angle-distance space—typically angle-range space—traced by subcarriers or control parameters (Zhang et al., 18 Sep 2025). A terminological correction is necessary at the outset: “Joint Metrics Matter” does not define a metric called “Joint Angle-Distance” or “JAD trajectory”; it defines JADE as Joint Average Displacement Error, with no angle term in the metric definition (Weng et al., 2023).

1. Terminological scope and disciplinary usage

The most important encyclopedic fact about JAD trajectory is that the phrase is context-dependent. In some papers, especially in robotics, the phrase itself is absent, but the underlying construction is explicit: robot configuration is represented simultaneously by joint angles and a structured set of distances among rigidly attached points, or by mixed joint variables containing both revolute angles and prismatic lengths (Marić et al., 2020, Caro et al., 2015). In other papers, the phrase is explicit and central: the near-field localization paper defines a Joint Angle-Distance (JAD) trajectory as the locus of focusing points $(\theta_m,r_m)$ traced by subcarriers under a joint TTD+PS beamformer (Zhang et al., 18 Sep 2025). In FD-RIS communications, the dominant phrase is instead joint distance-angle beamforming, but the mathematical object is closely related: a controllable focal law in range-angle space (Xiao et al., 2024, Xiao et al., 24 May 2025, Xiao et al., 10 Nov 2025).

This terminological dispersion matters because identical words refer to different independent variables. In manipulator planning, the relevant independent variable is often time $t$ or path progress $s$, and the dependent variables are actuator coordinates $q(s)$ or $q(t)$. In wideband localization, the independent variable is frequently the subcarrier index $m$, and the dependent variables are beam focus coordinates $(\theta_m,r_m)$ (Fried et al., 2024, Zhang et al., 18 Sep 2025). A plausible implication is that “JAD trajectory” should be read less as a single canonical object and more as a family of hybrid angle-distance parameterizations whose precise meaning is domain-specific.

The most common misconception arises from confusion with JADE in multi-agent trajectory forecasting. That work studies joint displacement-based forecasting metrics, not angle-distance geometry; its central quantities are JADE, JFDE, and collision rate, and it explicitly does not define “Joint Angle-Distance” (Weng et al., 2023).

2. Distance geometry as a companion to joint-angle state

In robot kinematics, the strongest mathematical basis for a JAD formulation comes from distance geometry. “Inverse Kinematics as Low-Rank Euclidean Distance Matrix Completion” reformulates inverse kinematics by introducing a point set

$$\mathbf{P}=[\mathbf{p}_0,\mathbf{p}_1,\dots,\mathbf{p}_{N-1}]^\top \in \mathbb{R}^{N\times K}$$

and the associated EDM

$$\mathbf{D}=\mathcal{K}(\mathbf{X}),\qquad \mathbf{X}=\mathbf{P}\mathbf{P}^\top,$$

with entrywise relation

$D_{ij}=X_{ii}+X_{jj}-2X_{ij}=\|p_i-p_j\|^2.$

Because $t$0, inverse kinematics becomes a low-rank completion problem over a structured EDM rather than a search purely in angle space (Marić et al., 2020).

The corresponding optimization problem uses a mask $t$1 for known distances and a mask $t$2 for lower-bounded distances: $t$3 on the quotient manifold

$t$4

Known rigid geometry and target constraints populate $t$5, while symmetric joint-angle limits are encoded as lower bounds on selected distances (Marić et al., 2020).

“Riemannian Optimization for Distance-Geometric Inverse Kinematics” generalizes this viewpoint and formalizes the equivalence between distance-based IK and the distance geometry problem for a large class of articulated robots. It attaches points to joint axes, represents task constraints and symmetric joint limits in distance form, and optimizes over fixed-rank Gram matrices with a Riemannian trust-region method (Marić et al., 2021). The forward kinematic recursion remains angle-based,

$t$6

but the constraints become equalities or interval bounds on distances, including obstacle avoidance in terms of point-obstacle separations (Marić et al., 2021).

These papers do not define a temporal JAD trajectory directly. This suggests that their most rigorous contribution to JAD is a state representation: conventional joint variables $t$7 or $t$8 are paired with a geometric descriptor $t$9, $s$0, or $s$1. A plausible implication is that a time-indexed JAD trajectory in robotics would be a sequence such as $s$2 or $s$3, with temporal regularization added externally rather than supplied by the static IK formulation itself (Marić et al., 2020, Marić et al., 2021).

3. Path-parameterized and mixed-coordinate manipulator trajectories

A second major interpretation of JAD trajectory appears when the independent variable is path progression. “A Bi-Level Optimization Approach to Joint Trajectory Optimization for Redundant Manipulators” parameterizes each joint trajectory as a function of scalar path variable $s$4: $s$5 so that the full joint path is $s$6, with chain-rule derivatives

$s$7

The paper jointly optimizes the geometric joint path $s$8 and the timing law $s$9 through a bi-level structure: the lower level computes the fastest feasible path-speed profile for fixed $q(s)$0, and the upper level modifies $q(s)$1 itself under Cartesian path tolerance and joint-position constraints (Fried et al., 2024). In this sense, the paper realizes a literal joint-angle-versus-distance profile, where “distance” is the path parameter $q(s)$2.

The Orthoglide 5-axis paper supplies a more mechanical mixed-coordinate instance. The manipulator combines a 3-DOF translational parallel stage with a 2-DOF spherical wrist, so the actuator vector is

$q(s)$3

with two revolute joint angles $q(s)$4 and three prismatic joint lengths $q(s)$5 (Caro et al., 2015). The full trajectory is generated from task variables

$q(s)$6

through the inverse geometric model

$q(s)$7

and the rate mapping uses the inverse Jacobian

$q(s)$8

The paper does not name this a JAD trajectory, but in the Orthoglide context the joint trajectory is intrinsically mixed: the “distance” component is actuator displacement rather than a geometric EDM (Caro et al., 2015).

Taken together, these works show that JAD trajectory can refer either to a hybrid state augmentation by geometric distances or to a mixed actuation space containing both angles and linear displacements. The distinction is consequential: in the first case distance variables are redundant but geometrically expressive; in the second they are native actuator coordinates.

4. Per-frame JAD states in perception and human pose

Perception-oriented papers extend the same angle-plus-distance logic to estimation. “A Distance-Geometric Method for Recovering Robot Joint Angles From an RGB Image” constructs an EDM over joint-center points $q(s)$9, auxiliary axis points $q(t)$0, and base-frame anchor points, then learns a map

$q(t)$1

from a 2D EDM of detected keypoints to a 3D EDM. The predicted distance representation is converted to a point realization by classical multidimensional scaling, using

$q(t)$2

and then to joint angles by a deterministic geometric IK layer (Bilić et al., 2023). The paper formalizes each training sample as

$q(t)$3

This suggests a particularly clean framewise JAD state: joint angles and a complete geometric distance descriptor are paired explicitly, even though temporal dynamics are not modeled (Bilić et al., 2023).

An analogous structure appears in marker-free human pose refinement. “Joint angle model based learning to refine kinematic human pose estimation” derives 12 joint angles from 13 2D keypoints, stacks them into

$q(t)$4

and complements them with a limb-length matrix

$q(t)$5

while the base point is stabilized with a Savitzky-Golay filter and angle trajectories are refined with a two-layer BiGRU with attention (Peng et al., 15 Jul 2025). The temporal prior on each joint angle uses an 8th-order Fourier series,

$q(t)$6

The paper again does not formalize a JAD trajectory as such, but it supplies nearly all of its ingredients: articulation is carried by angle trajectories, while geometric consistency is carried by Euclidean distances between adjacent joints (Peng et al., 15 Jul 2025).

5. Explicit JAD trajectories in near-field localization and beamforming

The most literal arXiv use of “JAD trajectory” occurs in near-field communications. “Beam Squint Assisted Joint Angle-Distance Localization for Near-Field Communications” defines the JAD trajectory as the locus of near-field focusing points $q(t)$7 traced by OFDM subcarriers under a joint TTD+PS beamformer (Zhang et al., 18 Sep 2025). With start and end focusing points $q(t)$8 and $q(t)$9, the subcarrier-dependent trajectory is

$m$0

$m$1

This provides a one-to-one mapping from subcarrier index to spatial focal point and enables a coarse-to-fine estimator: a low-complexity coarse stage based on subcarrier power peaks, followed by local near-field MUSIC over multiple subcarriers with geometric averaging of the spectra (Zhang et al., 18 Sep 2025).

FD-RIS papers supply a closely related but more beamforming-oriented interpretation. “Frequency Diverse RIS (FD-RIS) Enhanced Wireless Communications via Joint Distance-Angle Beamforming” shows that time modulation generates harmonic components at $m$2, so the propagation phase contains both angle-dependent geometric terms and distance-dependent terms proportional to $m$3, thereby enabling joint distance-angle beamforming in far-field communication scenarios (Xiao et al., 2024). “Multi-Subarray FD-RIS Enhanced Multi-user Wireless Networks” strengthens that mechanism by assigning distinct modulation frequencies $m$4 to different subarrays and using time delays $m$5, so that the received-energy pattern depends jointly on distance, angle, subarray frequency offset, and delay phase (Xiao et al., 24 May 2025). “FD-RIS-Enhanced Covert Communications” makes the same point via its normalized beampattern and explicit phase-alignment condition

$m$6

where $m$7 contains the distance-dependent term $m$8 (Xiao et al., 10 Nov 2025).

These communication papers generally speak of joint distance-angle beamforming rather than JAD trajectory. This suggests that the closest “trajectory” object in them is the beam-peak locus or high-gain ridge in angle-range space, indexed by subcarrier or by modulation parameters rather than by physical time.

6. Probabilistic geometry, misconceptions, and scope limits

A final clarification is provided by “Joint Distribution of Distance and Angles in Finite Wireless Networks.” That paper does not study trajectories, but it gives an exact joint law for distance and azimuth angle in finite regions with an arbitrarily placed reference node, showing that the two variables are generally correlated (Martín-Vega et al., 2022). For a convex region, the key geometric quantity is the boundary radius $m$9, and the support is angle-dependent: $(\theta_m,r_m)$0 In a rectangular region, the joint PDF becomes

$(\theta_m,r_m)$1

with admissibility determined by piecewise angular intervals that depend on $(\theta_m,r_m)$2 and on the reference-node offset (Martín-Vega et al., 2022). A plausible implication is that any stochastic JAD trajectory model in finite domains should inherit this geometry-induced dependence, rather than assuming independent angle and distance coordinates.

Several scope limits follow from the literature. First, JADE in multi-agent forecasting is unrelated to JAD and should not be cited as an angle-distance metric (Weng et al., 2023). Second, the distance-geometric IK papers are fundamentally single-pose formulations; trajectory smoothness, branch continuity, and dynamics must be added separately (Marić et al., 2020, Marić et al., 2021). Third, the near-field localization paper provides an explicit JAD trajectory, but its state space is beam focus $(\theta_m,r_m)$3, not manipulator joint space (Zhang et al., 18 Sep 2025). Fourth, the human-pose and monocular-robot-estimation papers provide strong per-frame angle-plus-distance representations, yet temporal coupling is either learned as a denoising prior or left implicit (Peng et al., 15 Jul 2025, Bilić et al., 2023).

Across these uses, the most stable encyclopedic interpretation is therefore not a single formula but a structural principle: a JAD trajectory couples articulation or steering angles with distance-valued geometry, and its independent variable may be time, path progress, subcarrier index, or control iteration depending on the field.