MOMENTUM CONSERVATION IN PHYSICS ENGINES

DETAILED ACTION Claims 1 – 20 have been presented for examination. Claims 1, 5, 9, 11, 13 and 17 are currently amended. This office action is in response to the amendments dated 07/21/2026. Response to Claim Objection Applicant’s amendments overcome the claim objection. Therefore, it is withdrawn. Response to Claims 1-20 define patentable subject matter under 35 U.S.C. § 101 Applicant’s arguments have been fully considered. However, the Office does not consider them to be persuasive. Applicant argues: “The Office Action characterizes the claims as merely reciting "split forces," "determine," and "apply" operations that allegedly may be performed mentally or with pen and paper. Applicant respectfully submits that this characterization overlooks the claim as a whole and ignores the technological context in which the claimed operations are performed. The claims are not directed to merely evaluating information or determining mathematical relationships in the abstract. Rather, the claims are directed to simulation of articulated-body movement using momentum-based and center-of-mass-based correction operations that modify movement of links of an articulated body within a simulation … Thus, the claims are directed to operation of a computer-implemented articulated-body simulation framework and not to a process that can practically be performed in the human mind.” (emphasis added) Applicant argues, in part, that the claims are “within a simulation”. Examiner notes that the “simulate” explicitly recites mathematical features and relationships to effectuate the simulation (see emphasis above). Therefore, it is part of the abstract idea and does not by itself transform the claimed invention into eligible subject matter. Further, the only additional elements of claim 1 are “processing units” which does not amount to integrating the abstract idea steps into a practical application, and does not amount to significantly more than the abstract idea. Applicant argues: “Indeed, there is no reasonable way that a human, either mentally or using pencil and paper, could practically perform the claimed operations of determining momentum values from integrated force domains, determining a velocity from a momentum discrepancy, determining a center-of mass difference vector using a previous center-of-mass position determined from center-of-mass velocity information, and simulating movement of an articulated body having multiple links by applying velocity corrections to the links based on the vector. Such operations require maintenance and modification of articulated-body simulation state and are fundamentally computer-implemented simulation operations … Accordingly, the claim is directed to a specific application of momentum-based and center-of-mass-based correction techniques within an articulated-body simulation environment.” (emphasis added) Applicant argues that the claimed steps are directly linked to “maintenance and modification of articulated-body simulation state” (see emphasis above). Examiner notes the “simulate” covers intangible application of the velocity and vector since it is limited only by the recited mathematical features and relationships. Therefore, the limitation when considered individually does not require a computer-implementation. Further, implementation of the steps with generic “processing units” does not integrate the abstract idea into a practical application, nor does it amount to significantly more than the abstract idea. Applicant argues: “Applicant further respectfully submits that amended claim 1 improves the technological field of articulated-body simulation … Thus, the claim is directed to improving operation of articulated-body simulation systems rather than merely performing mathematical calculations” Examiner notes that limiting the claimed invention to a field of use does not integrate the abstract idea into a practical application, nor does it amount to significantly more than the abstract idea. Applicant argues: “Applicant submits that amended claim 1 recites a specific set of machine-implemented operations for simulating movement of an articulated body and is not directed to merely evaluating information or performing mental reasoning in the abstract … The specification explains that articulated-body simulations may become unstable due to momentum drift and that internal forces should not change overall momentum while external forces may change momentum … The specification further explains that center-of-mass-based correction operations may be performed to address linearization artifacts associated with articulated-body simulation” Applicant argues various benefits which flow from the abstract idea itself. Therefore, Applicant’s arguments are not persuasive. Applicant argues: “Rather, amended claim 1 is directed to computer Implemented articulated-body simulation and recites a specific set of machine-implemented operations that operate on simulation state information associated with an articulated body having multiple links and joints … The October 2019 Guidance provides examples of operations that cannot practically be performed in the human mind because they involve specialized computational processing. Much like those examples, there is no reasonable way that a human, either mentally or using pencil and paper, could practically perform the claimed operations … These operations require maintenance and modification of articulated-body simulation state and are performed within a computer-implemented simulation framework … Amended claim 1 does not merely recite determining mathematical relationships. Rather, amended claim 1 recites a specific machine-implemented workflow in which … Rather, amended claim 1 expressly recites simulating movement of an articulated body by applying velocity corrections to one or more links based on a vector corresponding to a difference between center-of-mass positions … Thus, the claim applies any alleged mathematical concepts in a specific technological context and uses the resulting information to improve operation of an articulated-body simulation system. The practical application of amended claim 1 is therefore directed toward improving computer-implemented articulated-body simulation through momentum-based and center-of mass-based correction of simulated articulated-body movement” Applicant’s arguments are not persuasive based on the preceding remarks. Applicant argues: “Applicant respectfully submits that amended claim 1 recites a set of operations that improves the technical field of computer-implemented articulated-body simulation and physics engine processing … Amended claim 1 reflects these disclosed improvements … Accordingly, amended claim 1 is not directed merely to determining mathematical relationships. Rather, the claim recites a specific technological solution to a technological problem in articulated-body simulation” Applicant’s arguments are not persuasive based on the preceding remarks. Specifically, limiting the claimed invention to a field of use does not integrate the abstract idea into a practical application, nor does it amount to significantly more than the abstract idea. Further, the benefits flow wholly from the abstract idea steps, and the computer implementation is recited at a high-level of generality. Response to Claims 1-4, 6-12, 14-16, and 19-20 are allowable under 35 U.S.C. § 103 over Nenchev and Wang Applicant’s amendments overcome the prior art rejection. Therefore, it is withdrawn. Applicant argues: “Amended claim 1 now incorporates the essence of this previously indicated allowable subject matter. In particular, amended claim 1 recites determining a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, wherein the previous center-of-mass position is determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body, and further recites simulating movement of the articulated body by applying velocity to one or more links of the articulated body based at least on the vector … In contrast, the present specification explains that center-of-mass-based correction operations are performed to address linearization artifacts associated with articulated-body simulation … Thus, the claimed vector is not merely calculated. Rather, the vector is used to modify movement of links of the articulated body during simulation. The amended claim expressly recites simulating movement of the articulated body by applying velocity to one or more links of the articulated body based at least on the vector … Applicant respectfully submits that claim 9 is allowable at least for reasons including some of those discussed above in connection with claim 1 … Applicant respectfully submits that claim 17 is allowable at least for reasons including some of those discussed above in connection with claim 1. For example, claim 1 [sic] recites, in part, the features of … [claim 17 amended claim language]” Applicant has substantially rolled up the allowable subject matter from claim 5 into the independent claims covering specific mathematical constructs and features, with the exception of the performance of the limitations “after the integrating the one or more external forces and the one or more internal forces”. Allowable Subject Matter The following is a statement of reasons for the indication of allowable subject matter, subject to overcoming the 101 rejection: None of the prior art of record taken individually or in combination discloses the claim 1 (and similarly claim 9, and by incorporation claims 2 – 8 and 10 - 16) “determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; and simulate movement of the articulated body by applying the velocity to one or more links of the articulated body based at least on the vector”, in combination with the remaining elements and features of the claim. It is for these reasons that the applicant’s invention defines over the prior art of record. None of the prior art of record taken individually or in combination discloses the claim 17 (by incorporation claims 18 – 20) “simulate an articulated body based at least on an adjustment to a velocity that is associated with a root link of the articulated body, the adjustment to the velocity being computed based at least on constraining a change in momentum determined from one or more internal forces associated with the articulated body using a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body”, in combination with the remaining elements and features of the claim. It is for these reasons that the applicant’s invention defines over the prior art of record. He et al. “Biped Robot Walking Control with Centrodial Angular Momentum Preview” teaches compensating the centroidal moment pivot trajectory center of mass with angular momentum. However, does not appear to explicitly disclose: determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; and simulate movement of the articulated body by applying the velocity to one or more links of the articulated body based at least on the vector. Xu et al. “Singularity-free Path Planning of Dual-arm Space Robot for Keeping the Base Inertially Stabilized during Target Capturing” teaches angular and linear momemta in combination with a center of mass. However, does not appear to explicitly disclose: determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; and simulate movement of the articulated body by applying the velocity to one or more links of the articulated body based at least on the vector. Resiman et a. “Coordination underlying the control of whole body momentum during sit-to-stand” teaches linear and angular momentum of the center of mass. However, does not appear to explicitly disclose: determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; and simulate movement of the articulated body by applying the velocity to one or more links of the articulated body based at least on the vector. Horsewill, I. "Lightweight Procedural Animation With Believable Physical Interactions" teaches a physically plausible way of animating articulated bodies. However, does not appear to explicitly disclose: compute a linear velocity of the articulated body with respect to a center of mass of the articulated body after the integrating the one or more external forces and the one or more internal forces; determine, based at least on the linear velocity, a previous center of mass position prior to the integrating the one or more external forces and the one or more internal forces; determine a vector corresponding to a difference between the center of mass and the previous center of mass; and displace the one or more links based at least on the vector. Deits et al. (US 2022/0410378) teaches an expected motion of the center of mass of the robot computed by an MPC. However, does not appear to explicitly disclose: compute a linear velocity of the articulated body with respect to a center of mass of the articulated body after the integrating the one or more external forces and the one or more internal forces; determine, based at least on the linear velocity, a previous center of mass position prior to the integrating the one or more external forces and the one or more internal forces; determine a vector corresponding to a difference between the center of mass and the previous center of mass; and displace the one or more links based at least on the vector. Any comments considered necessary by applicant must be submitted no later than the payment of the issue fee and, to avoid processing delays, should preferably accompany the issue fee. Such submissions should be clearly labeled “Comments on Statement of Reasons for Allowance.” Claim Rejections - 35 USC § 101 35 U.S.C. 101 reads as follows: Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title. Claims 1 – 20 are rejected under 35 U.S.C. 101 because the claimed invention is directed to a judicial exception (i.e., an abstract idea) without significantly more. Independent claim 1 recites at Step 1 a statutory category (i.e. a machine) system to: split forces acting on an articulated body into one or more external forces and one or more internal forces; determine, based at least on integrating the one or more external forces, a first momentum of the articulated body; determine, based at least on integrating the one or more internal forces, a second momentum of the articulated body; determine a velocity based at least on a delta between the first momentum and the second momentum; and determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; simulate movement of the articulated body by applying the velocity to one or more links of the articulated body. At Step 2A, Prong I the recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover performance of the limitations in the mind in combination with using a pen and paper (see MPEP 2106.04(a)(2)(III)). For example, the “split forces” and “determine … a first momentum” and “determine … a second momentum” and “determine a velocity” cover modeling actions recited at a high-level of generality in combination with a specific articulated body, which are reasonably performed in the mind. At Step 2A, Prong I the recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover mathematical concepts (see MPEP 2106.04(a)(2)(I)). The “determine a vector” and “simulate movement … by applying” covers intangible application of the velocity and vector since it is limited only by the recited mathematical features and relationships. Accordingly, the claim recites an abstract idea. At Step 2A, Prong II this judicial exception is not integrated into a practical application since the claimed invention further claims: one or more processing units to implement the steps. The “processing units” are recited at a high-level of generality such that they amount to no more than mere application of the judicial exception using generic computer components which does not amount to an improvement in computer functionality (see MPEP 2106.04(a)(I)). The claim is directed to an abstract idea. At Step 2B the claim does not recite additional elements that, alone or in an ordered combination, are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the recited “processing units” amounts to no more than mere instructions to apply the judicial exception using generic computer components. For at least these reasons, the claim is not patent eligible. Dependent claim 2 – 8 recite(s) at Step 1 the same statutory category as the parent claim(s), and further recite(s): Claim 2 compute an inverse of a compound inertia of the articulated body, further wherein the one or more processing units determine the velocity based at least on the inverse of the compound inertia; Claim 3 wherein the first momentum and the second momentum correspond to angular momentums, and the one or more processing units are further to compute a linear momentum based at least on at least one of the first momentum or the second momentum; Claim 4 wherein the velocity is applied to a root link of the one or more links; Claim 5 wherein the linear velocity of the articulated body with respect to the center of mass of the articulated body is being computed after the integrating the one or more external forces and the one or more internal forces, and wherein the one or more processing units are further to: determine, based at least on the linear velocity, the previous center of mass position prior to the integrating the one or more external forces and the one or more internal forces; determine the vector corresponding to a difference between the center of mass and the previous center of mass; and displace the one or more links based at least on the vector; Claim 6 determine a first compound inertia associated with the first momentum and a second compound inertia associated with the second momentum, wherein the delta corresponds to the first compound inertia and the second compound inertia. At Step 2A, Prong I the recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover mathematical concepts (see MPEP 2106.04(a)(2)(I)). For example, the “being computed after” covers mathematical calculations related to the articulated body since it further limits the parent claim “determine a vector” with regard to when the computation occurs. The recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover performance in the mind in combination with a piece of paper (see MPEP 2106.04(a)(2)(III)). For example, the “determine” cover to modeling actions recited at a high-level of generality in combination with a specific values. The “wherein the first momentum and the second momentum correspond to” and “wherein the velocity is applied to” and “wherein the delta corresponds to” further limit the parent claim abstract idea steps, without precluding performance in the mind. The “displace” covers intangible application of the velocity in a virtual environment (see claim 8). Accordingly, the claim(s) recite(s) an abstract idea. At Step 2A, Prong II this judicial exception is not integrated into a practical application since the claimed invention further claims: Claim 6 cause a presentation of the articulated body; Claim 8 wherein the system is comprised in at least one of: a system for performing simulation operations; a system for performing digital twin operations; a system for performing light transport simulation; a system for performing collaborative content creation for 3D assets; a system for performing deep learning operations; a system implemented using an edge device; a system implemented using a robot; a system for performing conservational AI operations; a system for real-time streaming applications; a system for presenting at least one of virtual reality content, augmented reality content, or mixed reality content; a system for generating synthetic data; a system incorporating one or more virtual machines (VMs ); a system implemented at least partially in a data center; or a system implemented at least partially using cloud computing resources. For example, the “cause a presentation” amounts to insignificant data outputting (see MPEP 2106.04(d)). The “system is comprised in at least one of” further limits the system to exist in the context with one or more commercially available computing systems, which amounts to no more than mere application of the judicial exception using generic computer components which does not amount to an improvement in computer functionality (see MPEP 2106.04(a)(I)). The claim is directed to an abstract idea. At Step 2B the claim(s) do not recite additional elements that, alone or in an ordered combination, are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the “system is comprised in at least one of” amounts to no more than mere instructions to apply the judicial exception using generic computer components. The “cause a presentation” covers well-understood, routine and conventional activity (see MPEP 2106.05(d)(II) “i. Receiving or transmitting data over a network”). Considering the additional elements in combination does not add anything more than when considering them individually since the “cause a presentation” requires no more than generic computer functions. For at least these reasons, the claim(s) are not patent eligible. Independent claim 9 recites at Step 1 a statutory category (i.e. a machine) processor to: split forces acting on an articulated body into one or more external forces and one or more internal forces; determine, based at least on integrating the one or more external forces, a first momentum of the articulated body; determine, based at least on integrating the one or more internal forces, a second momentum of the articulated body; determine a velocity based at least on a delta between the first momentum and the second momentum; determine a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body; simulate movement of the articulated body by applying the velocity to one or more links of the articulated body. At Step 2A, Prong I the recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover performance of the limitations in the mind in combination with using a pen and paper (see MPEP 2106.04(a)(2)(III)). For example, the “split forces” and “determine … a first momentum” and “determine … a second momentum” and “determine a velocity” cover modeling actions recited at a high-level of generality in combination with a specific articulated body, which are reasonably performed in the mind. At Step 2A, Prong I the recited limitations in part, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover mathematical concepts (see MPEP 2106.04(a)(2)(I)). The “determine a vector” and “simulate movement … by applying” covers intangible application of the velocity in a virtual environment limited only by the recited mathematical features and relationships. Accordingly, the claim recites an abstract idea. At Step 2A, Prong II this judicial exception is not integrated into a practical application since the claimed invention further claims: one or more processing units to implement the steps. The “processing units” are recited at a high-level of generality such that they amount to no more than mere application of the judicial exception using generic computer components which does not amount to an improvement in computer functionality (see MPEP 2106.04(a)(I)). The claim is directed to an abstract idea. At Step 2B the claim does not recite additional elements that, alone or in an ordered combination, are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the recited “processing units” amounts to no more than mere instructions to apply the judicial exception using generic computer components. For at least these reasons, the claim is not patent eligible. Dependent claims 10 – 16 are substantially identical to dependent claims 2 – 8, and are directed to abstract ideas for the same reasons. Independent claim 17 recites at Step 1 a statutory category (i.e. a machine) system to: simulate an articulated body based at least on an adjustment to a velocity that is associated with a root link of the articulated body, the adjustment to the velocity being computed based at least on constraining a change in momentum determined from one or more internal forces associated with the articulated body using a vector corresponding to a difference between a center of mass of the articulated body and a previous center-of-mass position associated with the articulated body, the previous center-of-mass position being determined based at least on a linear velocity of the articulated body with respect to the center of mass of the articulated body. At Step 2A, Prong I the recited limitations, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover mathematical concepts (see MPEP 2106.04(a)(2)(I)). For example, the “simulate” covers intangible application of the adjustment to the velocity which is “computed”, and the displacement which is “using a vector” since it is limited only by the recited mathematical features and relationships. Accordingly, the claim recites an abstract idea. At Step 2A, Prong II this judicial exception is not integrated into a practical application since the claimed invention further claims: one or more processing units to. The “processing units” are recited at a high-level of generality such that they amount to no more than mere application of the judicial exception using generic computer components which does not amount to an improvement in computer functionality (see MPEP 2106.04(a)(I)). The claim is directed to an abstract idea. At Step 2B the claim does not recite additional elements that, alone or in an ordered combination, are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the recited “processing units” amounts to no more than mere instructions to apply the judicial exception using generic computer components. For at least these reasons, the claim is not patent eligible. Dependent claim 18 – 20 recite(s) at Step 1 the same statutory category as the parent claim(s), and further recite(s): Claim 18 wherein the adjustment to the velocity is further computed based at least on a change in momentum determined from one or more external forces associated with the articulated body; Claim 19 wherein the one or more processing units determine the change in momentum by computing an angular momentum and computing a linear momentum using the angular momentum. At Step 2A, Prong I the recited limitations, alone or in combination, amount to steps that, under its broadest reasonable interpretation, cover mathematical concepts (see MPEP 2106.04(a)(2)(I)). For example, the “computed” and “computing” covers mathematical calculations related to the articulated body. Accordingly, the claim(s) recite(s) an abstract idea. At Step 2A, Prong II this judicial exception is not integrated into a practical application since the claimed invention further claims: Claim 20 wherein the system is comprised in at least one of: a system for performing simulation operations; a system for performing digital twin operations; a system for performing light transport simulation; a system for performing collaborative content creation for 3D assets; a system for performing deep learning operations; a system implemented using an edge device; a system implemented using a robot; a system for performing conservational AI operations; a system for real-time streaming applications; a system for presenting at least one of virtual reality content, augmented reality content, or mixed reality content; a system for generating synthetic data; a system incorporating one or more virtual machines (VMs ); a system implemented at least partially in a data center; or a system implemented at least partially using cloud computing resources. For example, the “system is comprised in at least one of” further limits the system to exist in the context with one or more commercially available computing systems, which amounts to no more than mere application of the judicial exception using generic computer components which does not amount to an improvement in computer functionality (see MPEP 2106.04(a)(I)). The claim is directed to an abstract idea. At Step 2B the claim(s) do not recite additional elements that, alone or in an ordered combination, are sufficient to amount to significantly more than the judicial exception. As discussed above with respect to the integration of the abstract idea into a practical application, the “system is comprised in at least one of” amounts to no more than mere instructions to apply the judicial exception using generic computer components. For at least these reasons, the claim(s) are not patent eligible. Conclusion THIS ACTION IS MADE FINAL. Applicant is reminded of the extension of time policy as set forth in 37 CFR 1.136(a). A shortened statutory period for reply to this final action is set to expire THREE MONTHS from the mailing date of this action. In the event a first reply is filed within TWO MONTHS of the mailing date of this final action and the advisory action is not mailed until after the end of the THREE-MONTH shortened statutory period, then the shortened statutory period will expire on the date the advisory action is mailed, and any nonprovisional extension fee (37 CFR 1.17(a)) pursuant to 37 CFR 1.136(a) will be calculated from the mailing date of the advisory action. In no event, however, will the statutory period for reply expire later than SIX MONTHS from the mailing date of this final action. Any inquiry concerning this communication or earlier communications from the examiner should be directed to ALFRED H. WECHSELBERGER whose telephone number is (571)272-8988. 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