{"id":"8bd8c34f-0ee3-4c05-b95a-7250deb10a45","arxiv_id":"2504.20615","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The authors extend invariant EKF and invariant smoother estimators for legged robots with LiDAR odometry and GPS observations, claiming better position accuracy than LIO-SAM and FAST-LIO2 on indoor and outdoor runs.","lead":"This robotics letter adapts invariant Kalman filtering and smoothing for quadrupeds to also consume LiDAR odometry and GPS, reporting reduced trajectory error on the HOUND2 robot. A reader might care because drift in robot position, especially vertical, is a practical bottleneck for legged robots working outdoors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equations (19)–(20) swap measurement and constant vector, so the LiDAR/GPS models do not satisfy the right-invariant form (7); absolute position is left-invariant, not right-invariant, so the claimed group-affine structure and convergence guarantees do not follow.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing flaw I find: the proposed LiDAR and GPS observations do not satisfy the right-invariant group-affine form (7) because the vector labeled 'constant' in Eqs. (19)–(20) actually contains the time-varying measured positions. This is not a peripheral typo; it is the central theoretical claim of the paper. If the models were instead expressed as left-invariant observations, the group-affine property could be recovered, and the practical algorithm might still be sound. But the paper explicitly claims right-invariance and builds the derivation of the filter and smoother on that claim. The swapped roles also make the Jacobians in Sec. III-D measurement-dependent, further indicating the derivation is internally inconsistent. The experimental results, which consist of single runs without code release or statistical comparison to the closest baseline (Leg-KILO), cannot compensate for the invalidity of the main contribution as stated. I therefore agree with the reader's REJECT verdict; a corrected theoretical derivation would be needed to reconsider.","tokens_in":14276,"tokens_out":9850,"duration_ms":99017,"concrete_test":"Analytical re-derivation: take the state X in (8) and an absolute position measurement p_t. Verify whether there exists a constant vector s such that p_t = X_t^{-1} s for all t; note that p_t is the position block of X_t, i.e., X_t s with constant s = [0,0,0,1]^T, which is a left-invariant output, not a right-invariant one. Then substitute the paper's definitions (19) into the E-IS residual r = X Y - b and Jacobian J = b^⊙ and check whether J is independent of the measurement. If the corrected left-invariant form yields the same update equations, the algorithm is salvageable, but the right-invariant claim in Sec. III is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central contribution of Sec. III is that LiDAR odometry and GPS can be integrated as right-invariant observations that preserve the group-affine property. However, Eqs. (19)–(20) state Ylid = X_t^{-1} blid and Ygps = X_t^{-1} bgps, with blid = [plid; 0; 1; 0] and bgps = [pgps; 0; 1; 0]. For the group-affine observation condition (7), the vector s must be constant. Here blid and bgps contain the measured positions plid and pgps, which vary at every step, so the models violate (7). The roles are also swapped: Ylid and Ygps are declared constant, so the 'observation' contains no measurement. An absolute position output is a left-invariant function of X (y = X s with constant s), not a right-invariant one y = X^{-1} s. Consequently, the log-linear error property and the convergence guarantees of [7] do not transfer to the model as written. The flaw propagates to Sec. III-D: the E-IS residual is r = X Y - b with Jacobian J = b^⊙; with b equal to the noisy measurement, J becomes measurement-dependent, which is inconsistent with the invariant-smoothing derivation. The experiments may still show that adding position corrections is practically useful, but the paper's stated theoretical contribution—preserving the invariant structure—is not established as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two state-estimation frameworks for quadruped robots, E-InEKF and E-IS, which fuse leg kinematics and IMU with LiDAR odometry (from KISS-ICP) and GPS position measurements. The central claim is that the LiDAR and GPS measurement models, given in Eqs. (19) and (20), satisfy the group-affine property of Eq. (7), thereby preserving the invariant-filtering and invariant-smoothing guarantees from Barrau and Bonnabel and from Yoon et al. The methods are evaluated on the KAIST HOUND2 robot in indoor and outdoor experiments against LIO-SAM, FAST-LIO2, P-InEKF, and P-IS, reporting lower ATE and RPE and reduced z-axis drift.","tokens_in":14478,"tokens_out":6512,"duration_ms":68023,"significance":"If the group-affine observation models were correctly derived, the paper would make a useful contribution: it would extend invariant filtering/smoothing for legged robots to include exteroceptive sensors while retaining the computationally attractive proprioceptive pipeline. The experimental study is also valuable, with comparisons against strong LiDAR-inertial baselines and an ablation of the LiDAR/GPS contributions. However, the theoretical centerpiece of the paper, the claim that Eqs. (19) and (20) are group-affine observations of the form of Eq. (7), is not correct as written; this must be fixed for the contribution to stand.","major_comments":[{"comment":"The LiDAR and GPS observation models are not in the group-affine form of Eq. (7). In Eq. (7), the observation y_t is the measurement (possibly a nonlinear function of the raw measurement) and s is a constant vector. In Eqs. (19) and (20), Y_lid and Y_gps are set to constant vectors, while b_lid and b_gps contain the time-varying measured positions p_lid and p_gps. This is exactly reversed: the measurement enters through s, the supposedly constant vector, and the observation variable is constant. Consequently, the models as written do not satisfy the right-invariant condition y_t = X_t^{-1} s + w_t with constant s, and the log-linear error and convergence guarantees of [7] do not transfer. An absolute position measurement is a left-invariant output of the state, y = X s (with s constant), not a right-invariant one; the paper provides no derivation showing how such a measurement can be cast in the right-invariant form while keeping s constant.","section":"Sec. III-B, Eqs. (19)-(20)"},{"comment":"The E-IS residual and Jacobian definitions inherit the same problem. The residual is r_lid = X Y_lid - b_lid, with b_lid containing the measured position p_lid, and the Jacobian is J_lid = [b_lid^⊙, 0]. Because b_lid is measurement-dependent, the Jacobian depends on the noisy measurement, which is inconsistent with the invariant-smoothing derivation in [12] where the Jacobian is evaluated at a constant vector. The measurement dependence enters the linearized cost and breaks the group-affine structure that the smoother is supposed to exploit. In addition, the text in Sec. III-D says \"plid ... represents the LiDAR position in the robot's body frame,\" which contradicts Sec. III-A where plid is the LiDAR position in the world frame; the paper must clarify which definition is used and reconcile it with the proposed residual.","section":"Sec. III-D"},{"comment":"The aggregated observation model of Eq. (21) does not rescue the formulation. Writing the observation as X_aug^{-1} b with b = [b_kin; b_lid; b_gps] is only valid if each b component is a constant; here b_lid and b_gps contain the measurements, so the right-hand side is not a linear function of a fixed vector. The update equations of Sec. II-C, which were derived for observations of the form of Eq. (7), therefore cannot be applied to this model. The Jacobians H_lid = H_gps = [0,0,-I_3,0] suggest a direct position measurement in world coordinates, which is a left-invariant observation, not the right-invariant form the paper claims. The paper needs to state which invariant structure it actually uses and re-derive the filter and smoother updates accordingly.","section":"Sec. III-C, Eq. (21)"}],"minor_comments":[{"comment":"The column headers for the outdoor results table are run together and missing spaces, making the table hard to read; please reformat.","section":"Table II"},{"comment":"The notation \"SEN+2(3)\" should be written as SE_{N+2}(3) to avoid confusion with SEN as a separate symbol.","section":"Sec. II-B"},{"comment":"The dimensions of Y_lid and b_lid are not explicitly stated, and it is unclear how the 5-vector [0_{3,1}; 0; 1; 0] is embedded in the higher-dimensional state X_t. This should be spelled out.","section":"Sec. III-B"},{"comment":"The statement that this is \"the first work to incorporate LiDAR odometry and GPS into InEKF and IS\" is broader than the evidence presented; please qualify the novelty claim against the existing invariant-filtering literature.","section":"Sec. I"}],"recommendation":"major_revision","confidential_remarks":"The reader's rejection is understandable because the core theoretical derivation is inconsistent as written. However, the defect is a correctable one: the authors can reformulate the LiDAR/GPS observations in a true invariant form (e.g., left-invariant position observations, or right-invariant body-frame measurements with a constant lever arm) and re-derive the filter and smoother updates. The experimental results may still be valid if the implementation uses the corrected form, so I do not judge the paper beyond repair. I recommend major_revision with a requirement that the authors resolve the measurement/constant-vector swap and clarify the invariant structure used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the hardware results are worth a look, but the theoretical core does not hold. Equations (19) and (20) are supposed to be right-invariant observations of the form y_t = X_t^{-1} s + w_t, with s constant. Instead, Y_lid and Y_gps are declared constant while b_lid and b_gps contain the measured positions. That is backwards. The measurement vector and the constant vector are swapped, and even after swapping, an absolute position in the world frame is a left-invariant output (y = X s), not a right-invariant one. The convergence guarantees from Barrau and Bonnabel therefore do not transfer. This is not a minor typo: it propagates into the E-IS residuals and Jacobians in Section III-D, so the central contribution—preserving the group-affine structure—is not established as written.\n\nWhat is genuinely new and useful: the paper is the first to bring LiDAR odometry and GPS into the invariant filter/smoother family for legged robots, and the experiments on the HOUND2 platform are non-trivial. The z-axis drift improvement over P-InEKF and P-IS is real, the ablations separating LiDAR and GPS contributions are helpful, and the filter-versus-smoother computation time comparison is a fair piece of engineering. The citation of prior invariant-filter work is accurate.\n\nSoft spots, in proportion: beyond the load-bearing derivation error, the experiments run each condition once, no code is released, the noise covariances are not disclosed, and the closest kinematically informed LiDAR method (Leg-KILO) is not compared. Those would matter even if the theory were sound. The practical idea may still work—a left-invariant or left-and-right-invariant observation model would be the natural fix—but the current manuscript does not support its stated contribution.\n\nFor a reading group, this is a useful case study in getting the invariant-observation bookkeeping right. I would not cite it in its current form, but it deserves a serious referee: the topic is important, the hardware data are real, and the flaw is correctable. Send it to peer review with the expectation of major revision.","headline":"The experimental results are plausible, but the LiDAR/GPS observation models in equations (19)-(20) swap the measurement and constant vector; an absolute position is left-invariant, not right-invariant, so the claimed group-affine guarantees do not hold.","tokens_in":15153,"tokens_out":4502,"would_cite":false,"duration_ms":46643,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that LiDAR odometry and GPS position readings can be written as right-invariant observations, letting a quadruped's invariant filter and smoother fuse them and cut position drift.","keywords":["sensor fusion","state estimation","quadruped robots","legged robots","invariant extended Kalman filter","invariant smoother","LiDAR odometry","GPS"],"falsifier":"Feed a recorded run back through the estimator with a fixed, time-invariant GPS position while the robot moves, and check whether the innovation and covariance evolve according to the state-independent linear equation (6); if they depend on the robot's pose, the observation is not group-affine and the central claim collapses.","tokens_in":13974,"feed_emoji":"🤖","tokens_out":15347,"duration_ms":129595,"temperature":0.7,"pith_summary":"Legged robots that estimate their state from leg kinematics and an IMU drift, especially along the vertical axis, because absolute position and yaw are unobservable. This paper tries to fix that by fusing LiDAR odometry and GPS into the invariant extended Kalman filter (InEKF) and invariant smoother (IS) frameworks that have become standard for legged state estimation. The authors' core claim is that a LiDAR or GPS position measurement can be cast in the right-invariant observation form $y = X^{-1}b + w$ with a constant vector $b$, which is exactly the condition that preserves the group-affine property and its state-independent error dynamics. They implement this on the HOUND2 quadruped with the LiDAR odometry computed in a parallel thread, and report that the fused estimators reduce position error, particularly z-drift, and beat LiDAR-inertial odometry baselines in indoor and outdoor tests. If true, the result matters because it brings drift-free global measurements into a theoretically principled fusion framework without sacrificing real-time efficiency.","feed_headline":"Quadruped position error drops up to 40% with fused LiDAR and GPS","feed_subtitle":"The filter keeps its state-independent error dynamics while folding in global position data.","key_machinery":"The machinery that carries the argument is the group-affine observation condition of equation (7), $y_t = X_t^{-1}s + w_t^{\\text{obs}}$ with $s$ constant, together with the proposed encodings of LiDAR and GPS positions as right-invariant observations: $Y_{\\text{lid}} = X^{-1}b_{\\text{lid}} + V_{\\text{lid}}$ and $Y_{\\text{gps}} = X^{-1}b_{\\text{gps}} + V_{\\text{gps}}$, where $b_{\\text{lid}} = [p_{\\text{lid}}, 0, 1, 0]^\\top$ and $b_{\\text{gps}} = [p_{\\text{gps}}, 0, 1, 0]^\\top$. These write absolute position output as if it were a right-invariant measurement, which is the format the invariant filter/smoother needs to keep the innovation a function of the invariant error alone. A supporting mechanical element is the parallel LiDAR thread that runs a point-to-point ICP algorithm to produce $p_{\\text{lid}}$ at roughly 10 Hz while the estimator thread continues at high rate, which is what keeps the added cost to about 0.06 ms per iteration for the filter.","core_discovery":"On its own terms, the paper's discovery is that global position measurements—the output of a LiDAR odometry system and a GPS receiver—can be expressed as right-invariant observations of the form $Y = X^{-1}b + V$, with $b$ a vector that packs the measured position together with zeros and ones, as in equations (19) and (20). Because the observation inherits the right-invariant form of equation (7), the authors argue, the log-linear error dynamics of the invariant filter and smoother are preserved, so the fused estimators keep the convergence guarantees of the invariant filtering theory. The contact-aided kinematic observation is aggregated with these new LiDAR and GPS observations into a single update for the InEKF and a single residual for the IS, and the LiDAR odometry is produced by a point-to-point iterative closest point algorithm running in a separate thread so the main estimator loop stays fast. Experiments on the HOUND2 quadruped, indoors without GPS and outdoors with GPS, show reduced absolute trajectory error and relative position error compared to established LiDAR-inertial odometry systems and to the proprioceptive-only baselines, with the largest visible gain in the z-axis position.","pith_inferences":["The parallel-thread design suggests a generic recipe for invariant estimators: any odometry source that emits a position—camera, radar, or learned—could be fused without touching the group-affine core.","A direct testable extension is to compare the right-invariant encoding against a left-invariant or plain measurement model for absolute position; this would show whether the invariance structure itself drives the reported accuracy gains.","Since the GPS covariance is treated as fixed in the paper, modulating it with satellite geometry or dilution of precision is a natural next step that could further cut the outdoor error.","Because the smoother's window size controls both accuracy and computation time, an adaptive window that grows when the robot speed is low or when drift is detected could give the best of both filter and smoother."],"forward_implications":["The fused estimators cut position drift, especially along the z-axis, reporting up to 28% lower error indoors and 40% lower outdoors relative to two established LiDAR-inertial odometry systems.","The right-invariant encoding of absolute position keeps the estimator's error dynamics state-independent, preserving the convergence behaviour of the invariant filter and smoother.","The loosely coupled parallel LiDAR thread keeps the filter's computation at about 0.06 ms per iteration, while the smoother's cost grows with its window size, making the accuracy-versus-speed trade-off explicit.","Ablations show each exteroceptive source contributes: removing GPS increases the outdoor error, and removing LiDAR reverts to the larger drift of the proprioceptive-only configuration."],"supporting_citations":[{"why":"Supplies the group-affine property and the right-invariant observation condition (equation 7) that the proposed LiDAR and GPS models are claimed to satisfy.","marker":"[7]"},{"why":"The contact-aided invariant extended Kalman filter for legged robots that E-InEKF extends with LiDAR and GPS observations.","marker":"[8]"},{"why":"The invariant smoother for legged robots that E-IS extends; the residual and Jacobian structure for the smoother is built on it.","marker":"[12]"},{"why":"The point-to-point ICP LiDAR odometry that produces the position estimate fed into the proposed observation model.","marker":"[27]"},{"why":"A LiDAR-inertial odometry baseline against which the proposed methods' ATE and RPE are compared.","marker":"[1]"},{"why":"A second LiDAR-inertial odometry baseline used in the outdoor and indoor comparisons.","marker":"[3]"},{"why":"Establishes that absolute position is unobservable in proprioceptive-only legged state estimation, motivating the exteroceptive fusion.","marker":"[4]"},{"why":"The slip-rejection covariance adjustment used by the contact estimators in the proposed frameworks.","marker":"[6]"}],"fun_headline_variants":["Invariant fusion cuts quadruped drift up to 40%","LiDAR+GPS in invariant filter trims quadruped ATE by 40%","Quadruped position error reduced 40% via invariant fusion","Invariant smoother+GPS cuts quadruped drift up to 40%","Fusing LiDAR GPS and IMU in invariant filter cuts error 40%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a LiDAR or GPS position fix can be modeled as a right-invariant observation with a constant vector $b$; if the measured position entering $b$ makes the vector time-varying, the group-affine convergence guarantee does not apply.","fun_headline_variants_meta":{"raw":{"variants":["Invariant fusion cuts quadruped drift up to 40%","LiDAR+GPS in invariant filter trims quadruped ATE by 40%","Quadruped position error reduced 40% via invariant fusion","Invariant smoother+GPS cuts quadruped drift up to 40%","Fusing LiDAR GPS and IMU in invariant filter cuts error 40%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001683,"raw_usage":{"total_tokens":6702,"prompt_tokens":1009,"completion_tokens":5693,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":5596}},"tokens_in":625,"tokens_out":5693,"duration_ms":39435,"temperature":1.0,"reasoning_tokens":5596,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:25:35.592343+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Feed a recorded run back through the estimator with a fixed, time-invariant GPS position while the robot moves, and check whether the innovation and covariance evolve according to the state-independent linear equation (6); if they depend on the robot's pose, the observation is not group-affine and the central claim collapses.","supporting_citations":[{"cited_title":"The Invariant Extended Kalman Filter as a Stable Observer,","cited_arxiv_id":null,"evidence_quote":"Supplies the group-affine property and the right-invariant observation condition (equation 7) that the proposed LiDAR and GPS models are claimed to satisfy."},{"cited_title":"Contact-aided invariant extended Kalman filtering for robot state estimation,","cited_arxiv_id":null,"evidence_quote":"The contact-aided invariant extended Kalman filter for legged robots that E-InEKF extends with LiDAR and GPS observations."},{"cited_title":"Invariant smoother for legged robot state estimation with dynamic contact event information,","cited_arxiv_id":null,"evidence_quote":"The invariant smoother for legged robots that E-IS extends; the residual and Jacobian structure for the smoother is built on it."},{"cited_title":"KISS-ICP: In Defense of Point-to-Point ICP – Simple, Accurate, and Robust Registration If Done the Right Way,","cited_arxiv_id":null,"evidence_quote":"The point-to-point ICP LiDAR odometry that produces the position estimate fed into the proposed observation model."},{"cited_title":"LIO-SAM: Tightly-coupled lidar inertial odometry via smoothing and mapping,","cited_arxiv_id":null,"evidence_quote":"A LiDAR-inertial odometry baseline against which the proposed methods' ATE and RPE are compared."},{"cited_title":"FAST-LIO2: Fast direct LiDAR-inertial odometry,","cited_arxiv_id":null,"evidence_quote":"A second LiDAR-inertial odometry baseline used in the outdoor and indoor comparisons."},{"cited_title":"Legged robot state estimation with dynamic contact event information,","cited_arxiv_id":null,"evidence_quote":"The slip-rejection covariance adjustment used by the contact estimators in the proposed frameworks."}],"review_version":1}