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Communication-Free Obstacle Localization from Aggregate Wrench Measurements in Leader--Follower Cooperative Transport
Amin Kashiri, Aditya Mohan, Yasin Yazıcıoğlu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/7/2026, 7:46:12 AM
Summary
This paper presents a method for obstacle localization in a leader-follower robotic team transporting a rigid payload without explicit communication. The leader infers obstacle locations by measuring the aggregate wrench (force and torque) exerted by followers. By designing a specific follower control law, the relationship between payload motion (twist) and aggregate wrench becomes piecewise-linear. The leader uses an adaptive probing procedure to identify changes in this linear map, allowing it to recover the bearing, distance, and identity of the follower detecting each obstacle.
Entities (7)
Relation Signals (6)
Follower Robots → exerts → Aggregate Wrench
confidence 95% · The leader measures the followers' aggregate wrench, i.e., the combined force and torque they exert on the payload
Leader Robot → measures → Aggregate Wrench
confidence 95% · The leader measures the followers' aggregate wrench, i.e., the combined force and torque they exert on the payload
Follower Robots → reactsto → Obstacle
confidence 95% · follower robots assist and react to locally detected obstacles.
Adaptive Probing Procedure → enables → Obstacle Localization
confidence 93% · We give sufficient conditions for exact recovery... and develop an adaptive probing procedure... to obtain the required measurements.
Leader Robot → performs → Adaptive Probing Procedure
confidence 93% · develop an adaptive probing procedure in which the leader applies translational and rotational inputs
Obstacle → causes → Piecewise-Linear Relationship
confidence 90% · Each follower resists motion toward nearby obstacles, resulting in a piecewise-linear relationship between the payload's translational and angular velocity and the aggregate wrench.
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Abstract
Abstract:We consider obstacle localization for a team of robots cooperatively transporting a rigid payload without explicit inter-robot communication. A leader robot directs the payload's motion, while follower robots assist and react to locally detected obstacles. The leader measures the followers' aggregate wrench, i.e., the combined force and torque they exert on the payload, but cannot directly distinguish their individual reactions. We design a follower control law that allows the leader to recover obstacle locations from these measurements. Each follower resists motion toward nearby obstacles, resulting in a piecewise-linear relationship between the payload's translational and angular velocity and the aggregate wrench. Changes between adjacent linear regions reveal an obstacle's bearing and distance and identify the responding follower. We give sufficient conditions for exact recovery at a fixed payload configuration and develop an adaptive probing procedure in which the leader applies translational and rotational inputs to the payload to obtain the required measurements. We demonstrate the performance of the proposed method in simulations.
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- Source: https://arxiv.org/abs/2610.08324v1
- Canonical: https://arxiv.org/abs/2610.08324v1
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Communication-Free Obstacle Localization from Aggregate Wrench Measurements in Leader–Follower Cooperative Transport Amin Kashiri Aditya Mohan Yasin Yazıcıoğlu †thanks: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.†thanks: A. Kashiri is with the Department of Electrical and Computer Engineering at Northeastern University, Boston, MA (kashiri.a@northeastern.edu).†thanks: A. Mohan is with the Department of Electrical and Computer Engineering at Northeastern University, Boston, MA (lnu.un@northeastern.edu).†thanks: Y. Yazıcıoğlu is with the Departments of Mechanical and Industrial Engineering and Electrical and Computer Engineering at Northeastern University, Boston, MA (y.yazicioglu@northeastern.edu). Abstract We consider obstacle localization for a team of robots cooperatively transporting a rigid payload without explicit inter-robot communication. A leader robot directs the payload’s motion, while follower robots assist and react to locally detected obstacles. The leader measures the followers’ aggregate wrench, i.e., the combined force and torque they exert on the payload, but cannot directly distinguish their individual reactions. We design a follower control law that allows the leader to recover obstacle locations from these measurements. Each follower resists motion toward nearby obstacles, resulting in a piecewise-linear relationship between the payload’s translational and angular velocity and the aggregate wrench. Changes between adjacent linear regions reveal an obstacle’s bearing and distance and identify the responding follower. We give sufficient conditions for exact recovery at a fixed payload configuration and develop an adaptive probing procedure in which the leader applies translational and rotational inputs to the payload to obtain the required measurements. We demonstrate the performance of the proposed method in simulations. I Introduction When a team of robots carries a shared payload without communicating, the payload is their only coordination channel. In leader–follower schemes, a leader that knows the destination steers the payload while each follower reinforces the motion measured at its own grip, so the team transports a rigid body with no communication at all [1, 2]; the principle has been implemented on robots [3] and extended to flexible payloads [4]. Where communication or a shared model is available, transport is instead posed as decentralized optimization [5, 6, 7, 8] or adaptive control over a broadcast common state [9]. In these works, obstacle information is either absent or supplied directly to the robot that plans around it. A follower, however, may sense an obstacle that the leader cannot, and without communication it can convey this only through how it pushes on the payload. Such interaction has been used for inference: followers infer the motion intended by a leader that knows the obstacles [10], a leader recovers obstacles from the input of the single follower in a two-robot team [11], and inverse safety filtering infers constraints from the safety-filtered actions of other agents [12]. Each attributes the observed response to one robot. With several followers, the leader instead senses one aggregate wrench, three numbers regardless of team size, in which all reactions are summed, so no single measurement reveals which follower responded or to what. In this paper, we show that the leader can separate these reactions by actively probing the payload. We design a follower law under which each follower resists only motion carrying it toward a nearby obstacle. The map from payload twist to aggregate wrench then becomes piecewise linear, and the change between adjacent linear regions encodes one obstacle’s bearing and distance together with the follower that detected it. Recovering piecewise-linear maps from queries is well studied for hybrid systems [13] and ReLU networks [14]; here, the structure is designed into the team so that every region boundary has a physical meaning. Our contributions are: (i) a communication-free follower law that makes individual obstacle reactions separable in the aggregate wrench; (i) sufficient conditions under which measurements at a fixed payload configuration uniquely determine every detected obstacle and its responding follower; and (i) an adaptive probing procedure that selects the leader’s test motions to obtain these measurements, evaluated in simulation. I Problem Formulation I-A Setup We consider N robots transporting a rigid payload in the plane. Each robot maintains a rigid grip on the payload and can apply both force and torque. Robot 11 is the leader and alone knows the destination, while robots 2,…,N2,…,N are the followers. The followers do not know their world positions or the destination, and they do not observe other robots’ states or inputs. The robots do not explicitly communicate with each other. We express vectors in a world frame for the analysis, while each follower evaluates the control laws in its own grip frame, fixed to the payload. Let i r_i denote robot i’s grip offset from the payload’s center of mass, expressed in the world frame, and let i∈ℝ2 p_i ^2 denote the grip’s world position. Following the balanced-grip assumption in [15], we assume ∑i=2Ni= _i=2^N r_i= 0. This arrangement ensures that equal forces at the follower grips produce no net moment about the payload’s center of mass. The payload has mass M, moment of inertia J and viscous-friction coefficient μv _v. Its pose is =[c⊤,θ]⊤ x=[ p_c^\! ,\,θ]^\! , where c∈ℝ2 p_c ^2 is the position of its center of mass and θ∈ℝθ its heading. Let c=˙c v_c= p_c and ω=θ˙ω= θ denote its center-of-mass velocity and angular velocity, respectively. We collect these quantities into the payload twist, =[c⊤,ω]⊤=˙ ν=[ v_c^\! ,\,ω]^\! = x. Robot i applies a force i f_i and a torque τi _i at its grip. Each follower measures its grip velocity i v_i and the angular velocity ω, and knows its own grip offset, the team size N and the payload parameters. Obstacles are stationary disks of known radius. A follower detects an obstacle’s center location with respect to its grip when the distance to the obstacle’s center is less than dactd_ act. We index each detected obstacle–follower pair by a. Thus, an obstacle detected by multiple followers corresponds to multiple pairs. For pair a, let iai_a denote the follower, dad_a the distance from the obstacle center to the grip, and a n_a the unit vector pointing from the obstacle center to the grip. We call a n_a the bearing. A pair is active when 0<da<dact0<d_a<d_ act. The leader has the same local sensing and parameter knowledge as a follower. In addition, it knows all grip offsets, measures its own world pose and the payload’s linear and angular acceleration. Using its pose and the known grip geometry, the leader can compute the world positions of all grips. I-B System model The rigid payload connects motion measured at a grip to motion of the whole body. The velocity at grip i is the sum of the center-of-mass velocity and the velocity caused by rotation: i=c+ω⟂ v_i= v_c+ω r_i , where ⟂=[−riy,rix]⊤ r_i =[-r_iy,\,r_ix]^\! is the grip offset rotated by 90∘90 . We write this relation as i=Bi,Bi=[10−riy01rix]. v_i=B_i ν, B_i= bmatrix1&0&-r_iy\\ 0&1&r_ix bmatrix. (1) The same grip geometry determines how a local force acts on the payload. For planar vectors, we use ×=axby−aybx a× b=a_xb_y-a_yb_x to denote the scalar cross product. A force i f_i contributes the moment i×i r_i× f_i about the center of mass. Including the torque τi _i applied directly through the grip, robot i contributes the wrench ℱi=Bi⊤i+[00τi]⊤.F_i=B_i^\! f_i+ bmatrix0&0& _i bmatrix^\! . (2) Thus, BiB_i maps payload motion to grip velocity, while Bi⊤B_i^\! maps grip force to payload wrench. Following [15, 2], the payload dynamics are M˙c=∑i−μvc,Jω˙=∑i(τi+i×i)−μvJMω.M v_c= _i f_i- _v v_c, J ω= _i ( _i+ r_i× f_i )- _vJMω. (3) Each follower assists transport by compensating for 1/N1/N of the modeled viscous friction: icarry=μvc/N f_i carry= _v v_c/N and τicarry=μvJω/(NM) _i carry= _vJω/(NM). The follower obtains c v_c from its local measurements using c=i−ω⟂ v_c= v_i-ω r_i . To direct the team toward its destination, the leader selects a desired translation velocity d v_d and angular velocity ωd _d. For a desired twist d=[d⊤,ωd]⊤ ν_d=[ v_d^\! ,\, _d]^\! , the leader applies 1=K1(d−c),τ1=K2(ωd−ω)−1×1, f_1=K_1( v_d- v_c), _1=K_2( _d-ω)- r_1× f_1, (4) with positive gains K1K_1 and K2K_2. The last term of the torque cancels the moment produced by the leader’s own force. This controller is also used to generate probing motions for obstacle localization. Let ℱagg=∑i=2NℱiF agg= _i=2^NF_i denote the followers’ aggregate wrench. Summing (2) over all robots and using (3), the leader computes it from the measured accelerations and its own wrench: ℱagg=[M˙c+μvcJω˙+μvJω/M]−ℱ1.F agg= bmatrixM v_c+ _v v_c\\ J ω+ _vJω/M bmatrix-F_1. (5) Under the carrying law alone, the N−1N-1 followers apply a total force (N−1)μvc/N(N-1) _v v_c/N and a total grip torque (N−1)μvJω/(NM)(N-1) _vJω/(NM). Their carrying forces also produce moments about the center of mass, but those moments sum to zero because ∑i=2Ni= _i=2^N r_i= 0. The aggregate carrying wrench therefore is given by ℱagg=Ac,Ac=N−1Ndiag(μv,μv,μvJ/M).F agg=A_c ν, A_c= N-1Ndiag ( _v, _v, _vJ/M ). The matrix AcA_c contains no obstacle information. The next section adds an obstacle-dependent reaction and shows how the leader can recover obstacle locations from the resulting wrench. I Obstacle Recovery In this section, we design a follower reaction law and establish conditions under which the resulting aggregate wrench measurements uniquely determine obstacle locations and the responding followers. The analysis considers a fixed payload configuration, with the grip positions, obstacle distances dad_a, and bearings a n_a held fixed while the twist varies. Section IV then describes how the leader can collect the required measurements using finite-duration probing motions. I-A Follower reaction law Let iA_i be the set of active obstacle–follower pairs associated with follower i. In addition to its carrying force, follower i applies iavoid=∑a∈iβ(da)max(0,−a⊤i)a, f_i avoid= _a _iβ(d_a) \! (0,- n_a^\! v_i ) n_a, (6) where β:(0,dact)→(0,∞)β:(0,d_ act)→(0,∞) is a specified, strictly decreasing function known to all robots. The follower applies i=icarry+iavoid f_i= f_i carry+ f_i avoid, while its directly applied torque remains τicarry _i carry. Since a n_a points from the obstacle toward the follower, pair a reacts only when a⊤i<0 n_a^\! v_i<0, that is, when the follower moves toward the obstacle. The reaction force points away from the obstacle, grows with the approach speed and vanishes at rest. An active pair is resisting when the current motion makes its reaction nonzero. I-B Piecewise-linear map from twist to wrench By (1) and (2), an active pair a contributes ℱa()=βamax(0,−a⊤)a,a=Bia⊤a=[aia×a],F_a( ν)= _a \! (0,- g_a^\! ν ) g_a, g_a=B_i_a^\! n_a= bmatrix n_a\\ r_i_a× n_a bmatrix, (7) where βa=β(da) _a=β(d_a). The third entry of a g_a is hia(a)h_i_a( n_a), where hi()=i×h_i( n)= r_i× n denotes follower i’s signed moment arm in direction n. Summing over all active pairs gives ℱagg()=Ac+∑aℱa()F agg( ν)=A_c ν+ _aF_a( ν). The vector a g_a decides both when the pair resists, through a⊤=a⊤ia<0 g_a^\! ν= n_a^\! v_i_a<0, and the direction of its wrench. Positive scaling of the twist scales the wrench without changing which pairs resist. Pair a starts or stops resisting when a⊤ g_a^\! ν changes sign. The twists satisfying a⊤=0 g_a^\! ν=0 form a plane through the origin, which we call the pair’s switching plane. These planes divide twist space into regions called cells. Within a cell, every pair remains on the same side of its switching plane, so the set S of resisting pairs is fixed. For a twist in that cell, a resisting pair a contributes −βaaa⊤- _a g_a g_a^\! ν, while a nonresisting pair contributes zero. The aggregate wrench is therefore linear within the cell: ℱagg()=AS,AS=Ac−∑a∈Sβaaa⊤.F agg( ν)=A_S ν, A_S=A_c- _a∈ S _a g_a g_a^\! . (8) Each cell is therefore a cone from the origin. Fig. 1 draws the cells where these cones meet a sphere centered at the origin, each switching plane meeting it in a great circle. Fig. 1: The cells of twist space, which are cones, drawn where they meet a sphere centered at the origin. Each colored great circle is where one pair’s switching plane meets the sphere, and the shaded region is where one cell meets it. The black circle is the translation circle of Section IV-B, with crossings (triangles) and sample centers (circles), numbered as in Fig. 2(b), each surrounded by its three probes (crosses, spread exaggerated for visibility). Hidden parts are faded. I-C Recovery from observations An observation is a twist ν together with its aggregate wrench ℱagg()F agg( ν). Within a cell the map is linear by (8), so three observations whose twists 1,2,3 ν_1, ν_2, ν_3 lie in one cell S and are linearly independent, i.e., det[123]≠0 [ ν_1\ ν_2\ ν_3]≠ 0, determine the local matrix ASA_S of that cell: AS=[ℱagg(1)ℱagg(2)ℱagg(3)][123]−1.A_S= [F agg( ν_1)\ F agg( ν_2)\ F agg( ν_3) ] [ ν_1\ ν_2\ ν_3 ]^-1. (9) Two cells are adjacent if they lie on opposite sides of one switching plane and on the same side of every other. The leader does not know in advance which cell a twist lies in, or which cells are adjacent. For the exact-recovery result in Section I-D, we assume that measurements from two adjacent cells have been identified and grouped by cell, but the active pair separating these cells is unknown. Across the switching plane of pair a the wrench stays continuous, since the reaction vanishes there, but the local matrix gains or loses the term of a. If no other pair switches there, the matrix after the crossing minus the one before is a rank-one matrix: ΔA=Aafter−Abefore=∓βaaa⊤, A=A_ after-A_ before=∓ _a g_a g_a^\! , (10) with the negative sign when pair a starts resisting. Let λ be the nonzero eigenvalue of ΔA A and q a unit eigenvector. Then |λ|=βa‖a‖2|λ|= _a\| g_a\|^2 and =±a/∥a∥ q=± g_a/\| g_a\|, and since the first two entries of a g_a form the unit bearing, c=∥1:2∥=1/∥a∥c=\| q_1:2\|=1/\| g_a\|. It follows that a=±∥a∥=±/c,βa=|λ|c2,da=β−1(βa). g_a=±\| g_a\| q=± q/c, _a=|λ|c^2, d_a=β^-1( _a). (11) The sign of λ identifies the resisting side: If λ<0λ<0, pair a resists in the cell after the crossing; if λ>0λ>0, it resists in the cell before the crossing. We resolve the sign ambiguity in a g_a by choosing the candidate satisfying a⊤<0 g_a^\! ν<0 for an observed twist in the resisting cell. The first two entries of a g_a give the bearing a n_a. The responding follower iai_a is identified by matching the third entry of a g_a with the signed moment arms hi(a)h_i( n_a) of the followers. This match is unique under the conditions given in Section I-D. The obstacle center is then a=ia−daa o_a= p_i_a-d_a n_a. I-D Exact recovery guarantee The reconstruction of Section I-C needs two things: only pair a may switch between the two cells, and the recovered moment arm must single out its follower. The following assumptions ensure both. Assumptions. (A1) For every active pair a, no other follower has the same moment arm in the bearing direction a n_a: hi(a)≠hia(a)h_i( n_a)≠ h_i_a( n_a) for every i∈2,…,N∖iai∈\2,…,N\ \i_a\. (A2) No two distinct active pairs have parallel or opposite bearings: a≠±b n_a≠± n_b for a≠ba≠ b. Under these assumptions, three observations with linearly independent twists in each of two adjacent cells are enough to recover the pair whose switching plane separates them. Theorem 1 At a fixed pose, suppose (A1) and (A2) hold, and let the switching plane of an active pair a separate two adjacent cells, each containing three observations with linearly independent twists. Then the reconstruction of Section I-C, applied to the local matrices that (9) gives from these observations, recovers exactly the follower iai_a and the obstacle center a o_a. Proof: By (9), the observations in each of these cells determine its local matrix. Label these matrices A1A_1 and A2A_2. Since the first two entries of a g_a are a n_a, (A2) makes a g_a nonparallel to every other b g_b, so no other pair shares the switching plane of a, and only a changes its resisting status between the two cells. Hence A2−A1A_2-A_1 has the rank-one form in (10), with a negative nonzero eigenvalue if a resists in cell 2 and a positive one if it resists in cell 1. The reconstruction of Section I-C then recovers pair a, using an observed twist of the resisting cell to fix the sign of a g_a. Finally, (A1) makes the follower unique. IV The probing procedure This section presents a probing procedure that searches for the observations required by Theorem 1 and uses them to estimate obstacle locations. IV-A Probes, samples and local matrices A probe is a short out-and-back test motion about a fixed reference pose ref x_ ref. Before each probe, the leader re-centers the payload, which returns it close to ref x_ ref and reduces the motion left by the previous probe. It then probes with a twist p ν_p: it sets the desired twist in (4) to −p- ν_p for a time T, then to p ν_p, and records the twist and the aggregate wrench as the payload passes back near the reference position, so that every probe sees nearly the same map (Fig. 2(a)). Both steps are described in the Appendix. Algorithm 1 The probing procedure 1: 2: n0n_0: number of initial samples 3: ρ: spread of a sample 4: ϵr _r: residual tolerance 5: ϵD _D: jump tolerance 6: ϵm _m: merge tolerance 7: 8: C: candidates (i,)(i, o), follower i and obstacle center o 9: U: unresolved intervals 10: 11: sample at φ : sample of spread ρ centered at angle φ on the translation circle 12: A^(φ) A( ), r(φ)r( ), ¯(φ) u( ): its fitted matrix, residual and mean measured twist 13: EigDecomp(A)(A): eigenpairs (λl,l)( _l, q_l) of symmetric A, ‖l‖=1\| q_l\|=1, |λ1|≥|λ2|≥|λ3|| _1|≥| _2|≥| _3| 14: g, d d: estimates of a g_a and dad_a 15: hjh_j, j p_j: moment arm and grip position of follower j 16: ,←∅C,U← 17: sample at φk=2πk/n0 _k=2π k/n_0, k=0,…,n0−1k=0,…,n_0-1; φn0=2π _n_0=2π reuses the sample at φ0 _0 18: for k=0,…,n0−1k=0,…,n_0-1 do 19: Search(φk,φk+1 _k, _k+1) 20: remove (i′,′)(i , o ) from C if an earlier (i,)∈(i, o) has i=i′i=i and ‖−′‖≤ϵm\| o- o \|≤ _m 21: 22: procedure Search(φa,φb _a, _b) 23: if maxr(φa),r(φb)>ϵr \r( _a),r( _b)\> _r then 24: ←∪[φa,φb]U ∪\[ _a, _b]\ 25: else 26: D^←A^(φb)−A^(φa) D← A( _b)- A( _a) 27: (λl,l)l=13←( _l, q_l)_l=1^3← EigDecomp(D D) 28: if |λ2|≤ϵD<|λ1|| _2|≤ _D<| _1| then 29: c←∥(1)1:2∥c←\|( q_1)_1:2\|, β^←|λ1|c2 β←| _1|c^2 30: if c=0c=0 or β β not in the range of β then return 31: d^←β−1(β^) d←β^-1( β) 32: φ∗←φb ^*← _b if λ1<0 _1<0, else φa _a 33: ^←±1/c g←± q_1/c, with the sign giving ^⊤¯(φ∗)<0 g^\! u( ^*)<0 34: i←argminj|hj(^1:2)−g^3|i← _j|h_j( g_1:2)- g_3| 35: ←∪(i,i−d^^1:2)C ∪\(i,\, p_i- d\, g_1:2)\ 36: else if |λ2|>ϵD| _2|> _D then 37: if φb−φa≥2ρ _b- _a≥ 2ρ then 38: sample at φm=(φa+φb)/2 _m=( _a+ _b)/2 39: Search(φa,φm _a, _m), Search(φm,φb _m, _b) 40: else 41: ←∪[φa,φb]U ∪\[ _a, _b]\ (a) (b) (c) Fig. 2: (a) One probe, with the motion exaggerated: after re-centering, the payload moves out with −p- ν_p and back with p ν_p, and the twist and wrench are recorded as it passes back near the reference position. In this illustrative probe, the payload moved 2.32.3 m and turned 0.03∘0.03 , and the return ended 3.6μ3.6~ from the reference position. (b) Payload, leader (R1), followers (R2–R5) and the six sample-center directions for n0=6n_0=6. (c) One follower’s reaction (R3) along the translation circle, zero outside the arc between its crossings (dashed). The leader fits a matrix to each sample. Let j u_j and j w_j be the measured twist and wrench of probe j, both multiplied by the same factor so that all twists have the same magnitude. As shown in Section I-B, this rescaling does not change the local matrix. The fitted matrix is A^=argminA=A⊤∑j=02‖Aj−j‖2. A= A=A arg\,min _j=0^2\|A u_j- w_j\|^2. If the three twists lie in one cell, the fit returns its local matrix up to the measurement error, by (9). Its residual is r=(∑j‖A^j−j‖2)1/2r= ( _j\| A u_j- w_j\|^2 )^1/2. If the twists lie in different cells, a single symmetric matrix generally cannot fit them accurately, so a large residual indicates that the sample may straddle a switching plane. IV-B The procedure The procedure has five parameters: the number n0n_0 of initial samples, the spread ρ, and three tolerances ϵr _r, ϵD _D and ϵm _m (Algorithm 1). Samples. The leader places n0n_0 equally spaced sample centers on the translation circle, the pure-translation directions [cosφ,sinφ, 0]⊤[ ,\, ,\,0]^\! , φ∈[0,2π) ∈[0,2π) (line 17), shown in black in Fig. 1 (Fig. 2(b) shows n0=6n_0=6). Each pair’s switching plane meets this circle at two opposite directions, perpendicular to its bearing, called its crossings; Fig. 2(c) shows one pair’s reaction along the circle. A sample whose residual exceeds ϵr _r is not used, and the intervals next to it are reported unresolved (line 23). Comparing neighbors. For neighboring samples, the leader computes the jump D D, the difference of their fitted matrices (line 26), which by (10) sums the rank-one terms of the pairs that start or stop resisting between them. A nearly zero jump is treated as no pair switching, although a weak switch, or several switches whose terms cancel, can also produce one. A nearly rank-one jump is reconstructed as a single pair (line 28), and any other jump is split at its midpoint (line 38) until the interval is narrower than 2ρ2ρ, after which it is reported unresolved (line 41). Reconstruction. Lines 28–35 apply the reconstruction of Section I-C, with the follower whose moment arm is closest to the recovered one. If c=0c=0, or the recovered strength is not a value of β, the jump gives no candidate (line 30). Merging. Each pair is found at both of its crossings, so candidates that name the same follower with obstacle centers within ϵm _m of each other are merged (line 20). V Results V-A Setup We evaluate the procedure on two sets of scenes. The first contains five designed scenes, shown in Fig. 3. The second contains thirty random scenes, generated from fixed seeds, each with 3 to 6 followers at random but balanced grip positions around an ellipse with semi-axes drawn from 0.80.8–1.61.6 m and 0.50.5–1.21.2 m, and with the leader at a random interior point. Each follower receives an obstacle with probability 0.60.6, and then a second with probability 0.20.2, at a random bearing and a distance of 0.550.55–0.90.9 m, outside the payload and the robots. Every follower–obstacle pair closer than dactd_ act is active, which gives 147147 active pairs in total. Every run uses the same scenes. The simulations integrate (3) with n0=6n_0=6, the reaction strength β(d)=k1−d/dactβ(d)=k 1-d/d_ act, and the parameters of Table I. A pair is found when some estimate names its follower and lies within 55 cm of its obstacle, and an estimate that matches no pair is a false positive. Fig. 3: The designed scenes at the reference pose, with the number of active pairs. TABLE I: Parameters at K1=100K_1=100 N s/m. Some of these parameters scale with K1K_1, with the relation shown in the last column. Quantity At K1=100K_1=100 N s/m Scaling Payload Mass M, inertia J 22 kg, 0.50.5 kg m2 – Carrying gain μv _v 22 N s/m – Reaction Maximum strength k 11 N s/m – Activation distance dactd_ act 11 m – Leader Gain K2K_2 2525 N m s K1K_1 Probe Speed α 33 m/s – Hold T 0.20.2 s 1/K11/K_1 Re-centering duration 0.60.6 s 1/K11/K_1 Re-centering gain grg_r 1515 s-1 K1K_1 Integration step 88 ms 1/K11/K_1 Procedure Initial samples n0n_0 66 – Spread ρ 0.040.04 K1−1/4K_1^-1/4 Residual tolerance ϵr _r 3×10−63× 10^-6 1/K11/K_1 Jump tolerance ϵD _D 10−310^-3 K1−1/2K_1^-1/2 Merge tolerance ϵm _m 22 cm – TABLE I: Random scenes: pairs found and false positives at each leader gain K1K_1. K1K_1 (N s/m) 100100 10001000 10410^4 10510^5 10610^6 10810^8 Pairs found / 147147 5454 6868 8383 101101 120120 140140 False positives 33 00 00 11 00 00 TABLE I: Designed scenes: pairs found at each leader gain K1K_1, with no false positive. K1K_1 (N s/m) 100100 10001000 Single (1) 11 11 Basic (2) 22 22 Crowded (4) 44 44 Double (4) 44 44 Big (6) 44 66 Total (1717) 1515 1717 V-B Recovery At K1=100K_1=100 N s/m, the procedure finds 1515 of the 1717 designed-scene pairs, missing two in Big, and 5454 of the 147147 random-scene pairs, with three false positives. The probes are real motions, so each sample is recorded near, not exactly at, the reference pose, with twists that differ slightly from the commanded ones. To see how recovery improves as these deviations shrink, we raise K1K_1 up to 10810^8 N s/m and change the other parameters with it, following the empirically chosen schedule of Table I: shorter probes and faster re-centering keep the payload closer to the reference pose, a smaller spread makes it less likely that a sample straddles a switching plane, and smaller tolerances let weaker changes in the local matrices be detected. Recovery improves along this schedule. All 1717 designed-scene pairs are found at every gain from 10001000 N s/m (Table I), and random-scene recovery rises to 140140 of the 147147 pairs at 10810^8 N s/m, with one false positive at 10510^5 N s/m and none from 10610^6 N s/m (Table I). VI Conclusion In this work, we considered obstacle localization for a leader–follower team of robots cooperatively transporting a rigid payload without explicit inter-robot communication. Specifically, we investigated how the leader can localize obstacles outside its sensing range by measuring the aggregate wrench exerted on the payload by the followers. We proposed a reaction law under which each follower resists only motion toward nearby obstacles. We showed that this law makes the map from the payload’s twist to the followers’ aggregate wrench piecewise linear, with a rank-one change whenever a single follower-obstacle pair starts or stops resisting, and gave conditions under which each obstacle location and its responding follower are recovered in closed form from observations in adjacent cells. We also designed a probing procedure for the leader to search for such observations by comparing neighboring samples and splitting the intervals where several pairs switch. Simulation results demonstrated the performance of the proposed method. Future work will extend the proposed probing procedure to formally guarantee the collection of observations enabling exact recovery. We also intend to test our method on real robots. References [1] Z. Wang and M. Schwager (2016) Kinematic multi-robot manipulation with no communication using force feedback. In 2016 IEEE international conference on robotics and automation (ICRA), p. 427–432. 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Mironov (2020) Cryptanalytic extraction of neural network models. In Advances in Cryptology (CRYPTO), p. 189–218. Cited by: §I. [15] Z. Wang, G. Yang, X. Su, and M. Schwager (2018) Ouijabots: omnidirectional robots for cooperative object transport with rotation control using no communication. In Distributed Autonomous Robotic Systems: The 13th International Symposium, p. 117–131. Cited by: §I-A, §I-B. Appendix: implementation details Each sample consists of three probes. For a sample centered at angle φ with spread ρ, probe j=0,1,2j=0,1,2 uses the twist p,j=α+ρcosψj+ρsinψj31+ρ2,ψj=2πj3, ν_p,j=α\, c+ρ _j\, t+ρ _j\, e_3 1+ρ^2, _j= 2π j3, where =[cosφ,sinφ, 0]⊤ c=[ ,\, ,\,0]^\! is the sample center, the unit vectors =[−sinφ,cosφ, 0]⊤ t=[- ,\, ,\,0]^\! and 3=[0, 0, 1]⊤ e_3=[0,\,0,\,1]^\! span the directions around it, and the normalization gives every probe the speed α. Each probe takes three steps. First, the leader re-centers the payload: with ref=[c,ref⊤,θref]⊤ x_ ref=[ p_c, ref^\! ,\, _ ref]^\! , it sets the desired twist to −gr(−ref)-g_r( x- x_ ref) for a fixed duration, with a re-centering gain gr>0g_r>0. Second, it sets the desired twist to −p,j- ν_p,j for a time T, then to p,j ν_p,j. Third, during this return it records the twist and the aggregate wrench at the first instant at which (c−c,ref)⊤p,j≥0( p_c- p_c, ref)^\! v_p,j≥ 0, where p,j v_p,j is the translational part of p,j ν_p,j, so that every probe is recorded at nearly the same center-of-mass position.