Low-complexity human motion reconstruction method based on sparse inertial measurement unit

Abstract

The invention provides a low-complexity human body motion reconstruction method based on a sparse inertial measurement unit, and belongs to the technical field of virtual reality and three-dimensional human body motion reconstruction, and the method comprises the following steps: obtaining a data set required for reconstruction training based on human body postures, carrying out time modeling through a time sequence encoder, updating joint features on a human body skeleton graph, and obtaining a reconstruction result; a skeleton is divided into a trunk and four limbs according to a human anatomical structure, global rotation of a root joint and local rotation of each joint are respectively predicted by a partition kinematics regression head, low-rank decomposition is introduced into a large-scale linear layer to compress model parameters, and forward kinematics is utilized to recover three-dimensional joint positions of the whole body. In the training process, a two-stage teacher-student distillation model framework is adopted, a teacher network is trained through real labels, and then joint rotation and joint positions output by a teacher are used as soft targets to jointly restrain a student network through rotary distillation and position distillation. While the parameter quantity is reduced, the reconstruction precision and the motion smoothness of the whole body are improved.

Description

Technical Field

[0001] This invention relates to the field of virtual reality and 3D human motion reconstruction technology, and particularly to a low-complexity human motion reconstruction method based on sparse inertial measurement units (IMUs). This method employs a distillation-driven partitioned kinematic graph network structure to reconstruct the full-body human posture. Specifically, this invention addresses scenarios where only a small number of IMUs are integrated in terminals such as head-mounted displays and two-handed controllers. It utilizes pose observations estimated by these IMUs and their accompanying tracking systems to reconstruct the full-body 3D skeletal posture of the human body without the need for multi-camera or optical motion capture systems. Background Technology

[0002] With the rapid development of applications such as virtual reality (VR), augmented reality (AR), remote collaboration, and immersive gaming, accurately and stably reconstructing the user's full-body 3D posture in a typical home environment has become an important research direction in the field of human motion perception. Traditional motion capture systems typically rely on multi-camera arrays or optical marker devices. While these offer high accuracy, they are expensive, complex to deploy, and require specific shooting space and lighting conditions, making them unsuitable for everyday use and consumer-grade scenarios.

[0003] In contrast, sparse input schemes, based on a small number of inertial measurement units (IMUs) and their integrated terminal devices (such as head-mounted displays (HMDs) and handheld devices), offer advantages such as ease of use and low environmental dependence, and are gradually becoming an important research direction. However, these schemes can only acquire pose and motion information of some joints in the human body, and their observation dimensionality is far lower than the degrees of freedom of the complete human skeleton. How to recover high-precision whole-body skeletal pose from sparse observations is a highly ill-conditioned and multi-solution inverse problem.

[0004] In existing technologies, a typical approach is based on an optimization framework of inverse kinematics. This framework iteratively optimizes the pose of the entire human skeleton by taking into account pre-defined constraints on bone length, joint range of motion, and balance. However, this type of method heavily relies on manually designed energy functions and weights, making it prone to getting trapped in local optima. It is also sensitive to the initial pose and easily exhibits noticeable jitter and drift during rapid movement or under severe occlusion, making it difficult to maintain long-term stability in complex scenarios.

[0005] Another approach uses deep neural networks to directly regress whole-body pose from sparse sensor data, significantly improving inference speed and prediction accuracy. To better model the topological relationships between human joints, existing techniques have proposed graph convolutional skeletal networks. These networks treat the human skeleton as a graph structure with fixed connections, using graph convolutional networks to propagate features between joints, achieving explicit modeling of spatial structural information. Further work has introduced temporal convolutional or recurrent networks to jointly model long-term sequences, resulting in smoother temporal pose predictions.

[0006] However, existing graph convolution-based methods for full-body human pose reconstruction still have the following shortcomings: First, many networks treat all joints of the human body as homogeneous nodes and use a uniform graph convolution backbone for modeling, failing to fully utilize the differences in kinematic constraints, range of motion, and error distribution between the trunk (e.g., head, torso, pelvis) and limbs (upper limbs, lower limbs), especially in the lower limbs and end effectors (e.g., hands, feet), where large prediction errors are prone to occur; Second, some methods only supervise in the joint rotation angle space, lacking explicit constraints on the positions of key end joints, resulting in the need to correct positional deviations again in downstream applications, affecting the overall experience; Third, in order to improve expressive power, some graph networks introduce large-scale global attention modules and complex nonlocal operations, which can improve prediction accuracy to some extent, but also significantly increase the number of model parameters and computational overhead, making them unsuitable for deployment on embedded terminals or mobile devices.

[0007] Therefore, there is a need for a method for whole-body posture reconstruction that maintains high prediction accuracy, fully utilizes the kinematic differences between the human torso and limbs, applies explicit distillation constraints to key locations, and balances model parameter quantity and operational efficiency. To this end, this invention proposes a method for whole-body posture reconstruction using a distillation-driven partitioned kinematic graph network structure. Summary of the Invention

[0008] The purpose of this invention is to provide a low-complexity human motion reconstruction method based on sparse inertial measurement units, which solves the technical problems of existing whole-body posture reconstruction methods, which are prone to large joint posture reconstruction errors, large model parameter scale, and insufficient real-time performance when the number of sensors or computing resources are limited.

[0009] This invention proposes a distillation-driven partitioned kinematic graph network structure that comprehensively utilizes human skeletal topology and partitioned kinematic priors, significantly improving the accuracy and stability of human motion reconstruction while reducing the number of model parameters and computational complexity.

[0010] The sparse inertial measurement units are preferably deployed on key parts of the user's head, hands, etc., and can be worn independently or integrated into terminals such as head-mounted displays and handle fixing modules. They are used to collect acceleration, angular velocity, and position and orientation information of the corresponding parts through fusion estimation. This invention uses the pose sequence estimated by the sparse inertial measurement unit array and its matching tracking system as input to reconstruct the complete full-body skeletal posture.

[0011] A dataset suitable for human pose reconstruction training is acquired, and time-series encoders are used for modeling. Joint features are updated on the human skeleton map. Based on human anatomy, the skeleton is divided into trunk and limbs. A partitioned kinematic regression head predicts the global rotation of the root joints and the local rotation of each joint. Low-rank decomposition is introduced into a large-scale linear layer to compress model parameters. Finally, forward kinematics is used to recover the three-dimensional joint positions of the whole body. By introducing partitioned kinematic graph modeling and a two-stage teacher-student distillation training framework, the accuracy of whole-body pose reconstruction is effectively improved while reducing network complexity.

[0012] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0013] A low-complexity human motion reconstruction method based on sparse inertial measurement units, the method comprising the following steps:

[0014] Step 1: Obtain the dataset required for training based on human full-body posture reconstruction;

[0015] Step 2: Perform time modeling on the data to obtain a skeleton-level hidden feature sequence, and construct a skeleton diagram in the spatial dimension based on the topological relationship of the human skeleton, and propagate and aggregate joint features;

[0016] Step 3: Based on the human anatomical structure, the skeleton is divided into two categories: trunk and limbs. The regional kinematic regression head is used to predict the global rotation of the root joints and the local rotation of each joint. Low-rank decomposition is introduced into the large-scale linear layer to compress the model parameters. Then, forward kinematics is used to recover the three-dimensional joint positions of the whole body.

[0017] Step 4: Complete the reconstruction through a two-stage teacher-student distillation training.

[0018] Further, the specific process of step 1 is as follows: obtain the dataset for human motion reconstruction training and testing, select the subset of the publicly available human motion capture set AMASS for the whole body pose reconstruction task, the subset of the dataset includes the CMU dataset, BMLrub dataset and HDM05 dataset, the AMASS set performs unified skeleton topology and parameterization representation on each original motion capture library, provides the joint three-dimensional pose sequence in continuous time, and divide the samples in the subset of the dataset into training set and test set according to a predetermined ratio.

[0019] Furthermore, the ratio of the training set to the test set is 9:1.

[0020] Furthermore, the specific process of step 2 is as follows:

[0021] For each frame of pose observation obtained by the sparse inertial measurement units and their accompanying tracking system deployed on the head-mounted display and the two-handed controllers, the input sequence of length T is denoted as X = {x} t |t=1,2,…,T}, firstly, a temporal encoder is used to model X in time, resulting in a skeleton-level hidden feature sequence H={h t For |t=1,2,…,T}, in the spatial dimension, a skeleton graph G=(V,E) is constructed based on the topological relationships of the human skeleton, where V is the set of joints and E is the set of bone connections. Let A be the adjacency matrix of the graph and D be the degree matrix. For each graph convolutional unit, the joint features are propagated and aggregated using the following update method: Where W is the learnable weight matrix and σ(·) is the non-linear activation function, residual connections and normalization operations are introduced between several layers of graph convolution, so that the high-level joint features can enhance stability and expressive power while retaining low-level information.

[0022] Furthermore, the specific process of step 3 is as follows:

[0023] The skeleton features of frame t are divided into a torso feature set according to the joint index. With limb feature set Construct the trunk regression head f separately trunk (·) and the return of the limbs to the head f limb (·), to obtain the root joint rotation and local rotation parameters of each joint: Where q root,t Indicates root joint rotation, θ local,t For the local rotation parameter vectors of all joints in the body, in implementing f trunk (·) and f limb For large-scale linear mappings of (·), a low-rank decomposition method is used to decompose the weight matrix W into W≈UV, where the ranks of U and V are much smaller than the original matrix. This reduces the number of parameters and computational complexity while maintaining expressive power. Combined with the preset bone length parameter L, the three-dimensional positions of each joint are obtained by progressively accumulating transformations along the bone hierarchy using the forward kinematic function FK(·): p t =FK(q) root,t ,θ local,t ,L), where p t Let t be the set of three-dimensional positions of all joints in the whole body.

[0024] Furthermore, the specific process of step 4 is as follows:

[0025] First, a supervised loss is constructed based on the labeled data, and the predicted joint rotation is denoted as q. t The true value of joint rotation is Predict the overall pose or root pose as g t The truth value is the overall pose or root pose. The predicted three-dimensional position of the joint is p t The true value of the joint's three-dimensional position is The rotation supervision loss, global pose supervision loss, and position supervision loss are defined as follows:

[0026]

[0027] Where J is the number of joints, d rot (·,·) and d pose (·,·) denote the rotation and overall attitude metrics, respectively, and ||·||² denotes the Euclidean norm. Let the weighted supervision loss be... Where α rot α go α pos These are the weighting coefficients;

[0028] A two-stage teacher-student distillation mechanism is introduced. The first stage trains a high-capacity teacher network to obtain teacher outputs. In the second stage, on a student network with the same structure, rotary distillation loss, overall output distillation loss, and positional distillation loss are constructed:

[0029]

[0030] Let the total distillation loss be Where β rot β go β pos The distillation loss weights are used to define the total loss for the second stage of training as L. total =L sup +L kd By minimizing L total The student network was trained to approximate or outperform the teacher network and existing baseline methods in terms of joint rotation error, position error, and velocity error, while reducing the number of parameters and computational complexity.

[0031] The present invention, by adopting the above-described technical solution, has the following beneficial effects:

[0032] This invention employs a two-stage teacher-student distillation model framework during training. First, the teacher network is trained using real-world annotations. Then, the student network is jointly constrained by rotational and positional distillation, using the joint rotations and positions output by the teacher as soft targets. This invention improves the accuracy and smoothness of whole-body reconstruction while reducing the number of parameters. Attached Figure Description

[0033] Figure 1 This is an overall framework diagram of the implementation method of the present invention;

[0034] Figure 2 This is a schematic diagram of the skeleton diagram construction and partitioning of the present invention;

[0035] Figure 3 This is a diagram of the partitioned kinematic regression head and low-rank decomposition structure of the present invention;

[0036] Figure 4 This is a schematic diagram of the two-stage teacher-student distillation process of the present invention. Detailed Implementation

[0037] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and preferred embodiments. However, it should be noted that many details listed in the specification are merely to provide the reader with a thorough understanding of one or more aspects of the present invention, and these aspects of the invention can be implemented even without these specific details.

[0038] like Figure 1 As shown, the low-complexity human motion reconstruction method based on sparse inertial measurement units provided in this embodiment includes a temporal encoder, a skeleton graph convolutional backbone network, a regional kinematic regression head, a forward kinematics module, and a two-stage teacher-student distillation module. The overall process includes the following steps:

[0039] Step 1: Obtain the dataset required for training the full-body pose reconstruction. Select a subset of the publicly available human motion capture dataset AMASS for the full-body pose reconstruction task. This subset includes the CMU dataset (Lab, 2000), the BMLrub dataset (Troje, 2002), and the HDM05 dataset (Müller et al., 2007). The AMASS dataset provides a unified skeleton topology and parameterized representation for each original motion capture library, enabling the provision of continuous-time 3D joint pose sequences. Divide the samples in the subset of datasets into training and testing sets according to a predetermined ratio. Preferably, the ratio is 9:1, ensuring representative coverage of different motion categories in both sets during the partitioning process to facilitate evaluation of the generalization performance of the method.

[0040] Step 2: Perform time-series-skeleton joint modeling on the data. For example... Figure 2 As shown, this invention models the human skeleton in each frame as a graph structure. Figure 2 The left side shows a schematic diagram of a skeleton with 22 joint nodes, with the node numbers corresponding one-to-one with the joint indices in the actual implementation code; Figure 2The right side shows the corresponding adjacency matrix A, where black squares represent skeletal connections between joints, used to guide the propagation and aggregation of joint features in the graph convolutional backbone network. First, a temporal encoder is constructed to map the input to skeleton-level hidden features. For each frame t, the skeleton feature set is defined as follows: Where J represents the number of skeletal joints. Let be the feature vector of the j-th joint in frame t. The temporal encoder internally uses one-dimensional convolution to model local and long-range dependencies in the time dimension, and combines it with grouped embeddings divided by structures such as trunk, upper limb, and lower limb to learn independent linear transformations for different structural groups, thereby achieving structure-aware feature encoding.

[0041] After obtaining skeleton-level features, each frame's skeleton is modeled as a graph G = (V, E), V = {1, ..., J}, where the edge set E is constructed based on the human skeleton topology and its extended adjacency relationships. The skeleton graph convolutional backbone network uses multi-layer graph convolutions for node feature updates, and each layer can be represented as... in The input features are for the l-th layer. W is the normalized adjacency matrix. (l) σ is a learnable weight matrix, and σ(·) is a nonlinear activation function with residual connections and normalization. High-dimensional skeleton features incorporating spatiotemporal information are obtained through multi-layer stacking, providing input for subsequent partitioned kinematic regression.

[0042] Step 3: Regional kinematic regression and forward kinematic reconstruction. For example... Figure 3 As shown, the zoned kinematic regression head consists of two parts: the trunk regression head and the limb regression head, which act on the respective areas. Figure 2 The set of torso and limbs corresponding to the joint indexes. Figure 3 In the large-scale linear mapping layer, a low-rank decomposition structure is adopted, splitting the original weight matrix into product matrices with lower ranks to significantly compress the number of parameters while maintaining expressive power. Based on human kinematics, the joint index set V is divided into a trunk set Vt. trunk With limb set V limb and satisfy V trunk ∪V limb ={1,…,J}, Partitioning the skeleton features of frame t by index, we obtain Constructing a trunk regressive head f trunk (·) and limb return to head f limb (·), respectively predict the root joint pose and local rotation parameters: in For the root joint rotation matrix or quaternion, θ tThis represents the local rotational parameter vector for all joints in the body. Combined with the preset bone length parameter *l*, the three-dimensional positions of each joint are obtained layer by layer through the forward kinematic function *FK(·)*.

[0043] p j,t =FK j (θ t ,l),j=1,…,J

[0044] Step 4: Two-stage teacher-student distillation training and deployment. For example... Figure 4 As shown, this invention employs a two-stage teacher-student distillation framework: the first stage trains a high-capacity teacher network to approximate the real annotations in joint rotation, overall pose, and joint position space; the second stage, on a student network with the same structure or simplified parameters, uses the teacher output as a soft objective to construct three types of losses: rotational distillation, overall output distillation, and position distillation, thereby maintaining or even improving reconstruction accuracy while reducing model complexity. First, a supervised loss is constructed based on the labeled data. Given the ground truth joint rotation... Truth value: overall pose or root pose and true three-dimensional joint position Define the rotation supervision loss, global pose supervision loss, and position supervision loss as follows:

[0045]

[0046] Let the weighted surveillance loss be Where α rot α go α pos These are the weighting coefficients.

[0047] Building upon this, a two-stage teacher-student distillation mechanism is introduced. The first stage trains a high-capacity teacher network to obtain teacher outputs. The second stage involves constructing a distillation loss on student networks with identical or simplified structures:

[0048]

[0049] Total distillation loss is defined as Where λ rot , λ go , λ pos The distillation loss weighting coefficient is used. The total loss during the second phase of training is L. total =L sup +βL KD .

[0050] By minimizing L total While maintaining a low number of model parameters and low inference complexity, the student network can approach or even outperform the teacher network in terms of rotation error, position error, and velocity error.

[0051] VI. Experimental Results and Analysis

[0052] 1. Experimental conditions

[0053] In this embodiment, a subset of datasets for full-body pose reconstruction training is selected from the publicly available human motion capture dataset AMASS. The subset of datasets includes CMU (Lab 2000), BMLrub (Troje 2002), and HDM05 (Müller et al., 2007). The datasets are randomly divided into training and testing sets at a ratio of 9:1, which is consistent with the settings in the AvatarPoseEstimation open-source framework to ensure the comparability of the comparison results.

[0054] The experiment uses DKPG-Net as the student network model for this invention, and AvatarPoser is used to implement a high-capacity teacher network. Together, they form a teacher-student distillation framework. The input is the position and orientation sequence estimated by sparse inertial measurement units and their accompanying tracking systems deployed on the head-mounted display and the two-handed controllers. The output is the local rotation angle of each skeletal joint in each frame. Training and testing are both performed on a conventional GPU workstation (such as a deep learning server equipped with a single NVIDIA GPU) using the PyTorch deep learning framework.

[0055] To comprehensively assess the reconstruction quality, this embodiment uses three types of indicators consistent with existing work:

[0056] 1. MPJRE (Mean Per-Joint Rotation Error): The unit is degrees (°), which represents the average error between the predicted joint rotation and the actual rotation in axis-angle space;

[0057] 2. MPJPE (Mean Per-Joint Position Error): The unit is centimeters (cm), which represents the average Euclidean distance error of each joint position after mapping the rotation angle to Cartesian space through forward kinematics;

[0058] 3. MPJVE (Mean Per-Joint Velocity Error): The unit is centimeters per second (cm / s), which represents the average error between velocity vectors obtained by position difference between adjacent frames.

[0059] Among them, MPJVE is more sensitive to speed changes and can reflect whether the movement is smooth and natural. It is one of the important indicators for measuring the quality of whole-body posture reconstruction in a VR input system driven by sparse inertial measurement units deployed on head-mounted displays and hand controllers.

[0060] 2. Experimental Results

[0061] To verify the effectiveness of the method of this invention, DKPG-Net was compared with several existing offline or near real-time pose reconstruction methods, including representative methods such as Final IK, Cool Moves, LoBSTr, and AvatarPoser. All methods were evaluated under the same AMASS data partitioning and evaluation metrics. The comparison results are illustrated in Table 1.

[0062] Table 1 shows the comparison results.

[0063]

[0064] As shown in Table 1, under the same training data conditions, the proposed DKPG-Net outperforms existing comparative methods in all three metrics, especially exhibiting the smallest velocity error in MPJVE, indicating that this method can better recover the temporal continuity and dynamic smoothness of actions while ensuring spatial position accuracy. Compared with AvatarPoser, DKPG-Net achieves significant improvements in MPJRE, MPJPE, and MPJVE, demonstrating that the distillation-driven partitioned kinematic graph network structure can more fully utilize the prior knowledge of the teacher network while reducing network complexity.

[0065] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A low-complexity human motion reconstruction method based on sparse inertial measurement units, characterized in that, The method includes the following steps: Step 1: Obtain the dataset required for training based on human full-body posture reconstruction; Step 2: Perform time modeling on the data to obtain a skeleton-level hidden feature sequence, and construct a skeleton diagram in the spatial dimension based on the topological relationship of the human skeleton, and propagate and aggregate joint features; Step 3: Based on the human anatomical structure, the skeleton is divided into two categories: trunk and limbs. The regional kinematic regression head is used to predict the global rotation of the root joints and the local rotation of each joint. Low-rank decomposition is introduced into the large-scale linear layer to compress the model parameters. Then, forward kinematics is used to recover the three-dimensional joint positions of the whole body. Step 4: Complete the reconstruction through a two-stage teacher-student distillation training.

2. The low-complexity human motion reconstruction method based on sparse inertial measurement units according to claim 1, characterized in that, The specific process of step 1 is as follows: Obtain the dataset for human motion reconstruction training and testing, select the subset of the publicly available human motion capture set AMASS for the whole body pose reconstruction task, the subset includes the CMU dataset, BMLrub dataset and HDM05 dataset, AMASS performs unified skeleton topology and parameterization representation on each original motion capture library, provides three-dimensional pose sequences of joints in continuous time, and divides the samples in the subset into training set and test set according to a predetermined ratio.

3. The low-complexity human motion reconstruction method based on sparse inertial measurement units according to claim 2, characterized in that: The ratio of training set to test set is 9:

1.

4. The low-complexity human motion reconstruction method based on sparse inertial measurement units according to claim 1, characterized in that: The specific process of step 2 is as follows: For each frame of pose observation obtained by the sparse inertial measurement units and their accompanying tracking system deployed on the head-mounted display and the two-handed controllers, the input sequence of length T is denoted as X = {x} t |t=1,2,…,T}, firstly, a temporal encoder is used to model X in time, resulting in a skeleton-level hidden feature sequence H={h t For |t=1,2,…,T}, in the spatial dimension, a skeleton graph G=(V,E) is constructed based on the topological relationships of the human skeleton, where V is the set of joints and E is the set of bone connections. Let A be the adjacency matrix of the graph and D be the degree matrix. For each graph convolutional unit, the joint features are propagated and aggregated using the following update method: Where W is the learnable weight matrix and σ(·) is the non-linear activation function, residual connections and normalization operations are introduced between several layers of graph convolution, so that the high-level joint features can enhance stability and expressive power while retaining low-level information.

5. The low-complexity human motion reconstruction method based on sparse inertial measurement units according to claim 1, characterized in that, The specific process of step 3 is as follows: The skeleton features of frame t are divided into a torso feature set according to the joint index. With limb feature set Construct the trunk regression head f separately trunk (·) and the return of the limbs to the head f limb (·), to obtain the root joint rotation and local rotation parameters of each joint: Where q root,t Indicates root joint rotation, θ local,t For the local rotation parameter vectors of all joints in the body, in implementing f trunk (·) and f limb For large-scale linear mappings of (·), a low-rank decomposition method is used to decompose the weight matrix W into W≈UV, where the ranks of U and V are much smaller than the original matrix. This reduces the number of parameters and computational complexity while maintaining expressive power. Combined with the preset bone length parameter L, the three-dimensional positions of each joint are obtained by progressively accumulating transformations along the bone hierarchy using the forward kinematic function FK(·): p t =FK(q) root,t ,θ local,t ,L), where p t Let t be the set of three-dimensional positions of all joints in the whole body.

6. The low-complexity human motion reconstruction method based on sparse inertial measurement units according to claim 1, characterized in that, The specific process of step 4 is as follows: First, a supervised loss is constructed based on the labeled data, and the predicted joint rotation is denoted as q. t The true value of joint rotation is Predict the overall pose or root pose as g t The truth value is the overall pose or root pose. The predicted three-dimensional position of the joint is p t The true value of the joint's three-dimensional position is The rotation supervision loss, global pose supervision loss, and position supervision loss are defined as follows: Where J is the number of joints, d rot (·,·) and d pose (·,·) denote the rotation and overall attitude metrics, respectively, and ||·||² denotes the Euclidean norm. Let the weighted supervision loss be... Where α rot α go α pos These are the weighting coefficients; A two-stage teacher-student distillation mechanism is introduced. The first stage trains a high-capacity teacher network to obtain teacher outputs. In the second stage, on a student network with the same structure, rotary distillation loss, overall output distillation loss, and positional distillation loss are constructed: Let the total distillation loss be Where β rot β go β pos The distillation loss weights are used to define the total loss for the second stage of training as L. total =L sup +L kd By minimizing L total The student network was trained to approximate or outperform the teacher network and existing baseline methods in terms of joint rotation error, position error, and velocity error, while reducing the number of parameters and computational complexity.

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