A Least-Squares Estimation Approach to Improving the Precision of Inverse Dynamics Computations

1998 ◽  
Vol 120 (1) ◽  
pp. 148-159 ◽  
Author(s):  
A. D. Kuo
Keyword(s):  
Least Squares ◽  
Inverse Dynamics ◽  
Weighted Least Squares ◽  
Euler Method ◽  
Measured Data ◽  
Joint Torque ◽  
Ground Reaction Forces ◽  
Joint Torques ◽  
Wide Range ◽  
Reaction Forces

A least-squares approach to computing inverse dynamics is proposed. The method utilizes equations of motion for a multi-segment body, incorporating terms for ground reaction forces and torques. The resulting system is overdetermined at each point in time, because kinematic and force measurements outnumber unknown torques, and may be solved using weighted least squares to yield estimates of the joint torques and joint angular accelerations that best match measured data. An error analysis makes it possible to predict error magnitudes for both conventional and least-squares methods. A modification of the method also makes it possible to reject constant biases such as those arising from misalignment of force plate and kinematic measurement reference frames. A benchmark case is presented, which demonstrates reductions in joint torque errors on the order of 30 percent compared to the conventional Newton–Euler method, for a wide range of noise levels on measured data. The advantages over the Newton–Euler method include making best use of all available measurements, ability to function when less than a full complement of ground reaction forces is measured, suppression of residual torques acting on the top-most body segment, and the rejection of constant biases in data.

scholarly journals Joint Torque Prediction via Hybrid Neuromusculoskeletal Modelling during Gait Using Statistical Ground Reaction Estimates: An Exploratory Study

Sensors ◽  
2021 ◽  
Vol 21 (19) ◽  
pp. 6597
Author(s):  
Shui Kan Lam ◽  
Ivan Vujaklija
Keyword(s):  
Real World ◽  
Inverse Dynamics ◽  
Joint Torque ◽  
Force Measurements ◽  
Estimation Errors ◽  
Clinical Indicators ◽  
Musculoskeletal Modelling ◽  
Joint Torques ◽  
Biomechanical Models ◽  
Reaction Forces

Joint torques of lower extremity are important clinical indicators of gait capability. This parameter can be quantified via hybrid neuromusculoskeletal modelling that combines electromyography-driven modelling and static optimisation. The simulations rely on kinematics and external force measurements, for example, ground reaction forces (GRF) and the corresponding centres of pressure (COP), which are conventionally acquired using force plates. This bulky equipment, however, hinders gait analysis in real-world environments. While this portability issue could potentially be solved by estimating the parameters through machine learning, the effect of the estimation errors on joint torque prediction with biomechanical models remains to be investigated. This study first estimated GRF and COP through feedforward artificial neural networks, and then leveraged them to predict lower-limb sagittal joint torques via (i) inverse dynamics and (ii) hybrid modelling. The approach was evaluated on five healthy subjects, individually. The predicted torques were validated with the measured torques, showing that hip was the most sensitive whereas ankle was the most resistive to the GRF/COP estimates for both models, with average metrics values being 0.70 < R2 < 0.97 and 0.069 < RMSE < 0.15 (Nm/kg). This study demonstrated the feasibility of torque prediction based on personalised (neuro)musculoskeletal modelling using statistical ground reaction estimates, thus providing insights into potential real-world mobile joint torque quantification.

scholarly journals Simulation of Stand-to-Sit Biomechanics for Design of Lower-Limb Exoskeletons and Prostheses with Energy Regeneration

2019 ◽  
Author(s):  
Brock Laschowski ◽  
Reza Sharif Razavian ◽  
John McPhee
Keyword(s):  
Lower Limb ◽  
Electrical Energy ◽  
Mechanical Power ◽  
Future Research ◽  
Ground Reaction Forces ◽  
Dynamic Simulations ◽  
Inverse Dynamic ◽  
Joint Torques ◽  
Reaction Forces ◽  
Body Segment Parameters

AbstractAlthough regenerative actuators can extend the operating durations of robotic lower-limb exoskeletons and prostheses, these energy-efficient powertrains have been exclusively designed and evaluated for continuous level-ground walking.ObjectiveHere we analyzed the lower-limb joint mechanical power during stand-to-sit movements using inverse dynamic simulations to estimate the biomechanical energy available for electrical regeneration.MethodsNine subjects performed 20 sitting and standing movements while lower-limb kinematics and ground reaction forces were measured. Subject-specific body segment parameters were estimated using parameter identification, whereby differences in ground reaction forces and moments between the experimental measurements and inverse dynamic simulations were minimized. Joint mechanical power was calculated from net joint torques and rotational velocities and numerically integrated over time to determine joint biomechanical energy.ResultsThe hip produced the largest peak negative mechanical power (1.8 ± 0.5 W/kg), followed by the knee (0.8 ± 0.3 W/kg) and ankle (0.2 ± 0.1 W/kg). Negative mechanical work from the hip, knee, and ankle joints per stand-to-sit movement were 0.35 ± 0.06 J/kg, 0.15 ± 0.08 J/kg, and 0.02 ± 0.01 J/kg, respectively.Conclusion and SignificanceAssuming an 80-kg person and previously published regenerative actuator efficiencies (i.e., maximum 63%), robotic lower-limb exoskeletons and prostheses could theoretically regenerate ~26 Joules of total electrical energy while sitting down, compared to ~19 Joules per walking stride. Given that these regeneration performance calculations are based on healthy young adults, future research should include seniors and/or rehabilitation patients to better estimate the biomechanical energy available for electrical regeneration among individuals with mobility impairments.

Model-Based Design and Optimization of Passive Shoulder Exoskeletons

2021 ◽  
Author(s):  
Ali Nasr ◽  
Spencer Ferguson ◽  
John McPhee
Keyword(s):  
Elevation Angle ◽  
Differential Algebraic Equations ◽  
Joint Torque ◽  
Algebraic Equations ◽  
Joint Torques ◽  
Design And Optimization ◽  
Passive Mechanism ◽  
Wide Range ◽  
Passive Exoskeleton ◽  
Selection Of

Abstract To physically assist workers in reducing musculoskeletal strain or to develop motor skills for patients with neuromuscular disabilities, recent research has focused on Exoskeletons (Exos). Designing active Exos is challenging due to the complex human geometric structure, the human-Exoskeleton wrench interaction, the kinematic constraints, and the selection of power source characteristics. Because of the portable advantages of passive Exos, designing a passive shoulder mechanism has been studied here. The study concentrates on modeling a 3D multibody upper-limb human-Exoskeleton, developing a procedure of analyzing optimal assistive torque profiles, and optimizing the passive mechanism features for desired tasks. The optimization objective is minimizing the human joint torques. For simulating the complex closed-loop multibody dynamics, differential-algebraic equations (DAE)s of motion have been generated and solved. Three different tasks have been considered, which are common in industrial environments: object manipulation, over-head work, and static pointing. The resulting assistive Exoskeleton’s elevation joint torque profile could decrease the specific task’s human shoulder torque. Since the passive mechanism produces a specific torque for a given elevation angle, the Exoskeleton is not versatile or optimal for different dynamic tasks. We concluded that designing a passive Exoskeleton for a wide range of dynamic applications is impossible. We hypothesize that augmenting an actuator to the mechanism can provide the necessary adjustment torque and versatility for multiple tasks.

scholarly journals Inverse and forward dynamics: models of multi–body systems

2003 ◽  
Vol 358 (1437) ◽  
pp. 1493-1500 ◽  
Author(s):  
E. Otten
Keyword(s):  
Inverse Dynamics ◽  
Temporal Patterns ◽  
Forward Dynamics ◽  
Muscle Forces ◽  
Complex Behaviour ◽  
Joint Torques ◽  
Advantages And Disadvantages ◽  
Reaction Forces ◽  
Internal Forces ◽  
Multi Body

Connected multi–body systems exhibit notoriously complex behaviour when driven by external and internal forces and torques. The problem of reconstructing the internal forces and/or torques from the movements and known external forces is called the ‘inverse dynamics problem’, whereas calculating motion from known internal forces and/or torques and resulting reaction forces is called the ‘forward dynamics problem’. When stepping forward to cross the street, people use muscle forces that generate angular accelerations of their body segments and, by virtue of reaction forces from the street, a forward acceleration of the centre of mass of their body. Inverse dynamics calculations applied to a set of motion data from such an event can teach us how temporal patterns of joint torques were responsible for the observed motion. In forward dynamics calculations we may attempt to create motion from such temporal patterns, which is extremely difficult, because of the complex mechanical linkage along the chains forming the multi–body system. To understand, predict and sometimes control multi–body systems, we may want to have mathematical expressions for them. The Newton–Euler, Lagrangian and Featherstone approaches have their advantages and disadvantages. The simulation of collisions and the inclusion of muscle forces or other internal forces are discussed. Also, the possibility to perform a mixed inverse and forward dynamics calculation are dealt with. The use and limitations of these approaches form the conclusion.

scholarly journals Single Task Optimization-Based Planar Box Delivery Motion Simulation and Experimental Validation

2021 ◽  
Vol 13 (2) ◽  
Author(s):  
Yujiang Xiang ◽  
Shadman Tahmid ◽  
Paul Owens ◽  
James Yang
Keyword(s):  
Joint Angle ◽  
Inverse Dynamics ◽  
Optimization Method ◽  
Joint Torque ◽  
Motion Simulation ◽  
Single Task ◽  
Joint Torques ◽  
Time Integral ◽  
Design Variables ◽  
Complicated Task

Abstract Box delivery is a complicated task and it is challenging to predict the box delivery motion associated with the box weight, delivering speed, and location. This paper presents a single task-based inverse dynamics optimization method for determining the planar symmetric optimal box delivery motion (multi-task jobs). The design variables are cubic B-spline control points of joint angle profiles. The objective function is dynamic effort, i.e., the time integral of the square of all normalized joint torques. The optimization problem includes various constraints. Joint angle profiles are validated through experimental results using root-mean-square-error (RMSE) and Pearson’s correlation coefficient. This research provides a practical guidance to prevent injury risks in joint torque space for workers who lift and deliver heavy objects in their daily jobs.

Torque patterns of the limbs of small therian mammals during locomotion on flat ground

2002 ◽  
Vol 205 (9) ◽  
pp. 1339-1353 ◽  
Author(s):  
Hartmut Witte ◽  
Jutta Biltzinger ◽  
Rémi Hackert ◽  
Nadja Schilling ◽  
Manuela Schmidt ◽  
...  
Keyword(s):  
Ankle Joint ◽  
Hip Joint ◽  
Time Course ◽  
Wrist Joint ◽  
Ground Reaction Forces ◽  
Inverse Dynamic ◽  
Scanning Method ◽  
Joint Torques ◽  
Joint Forces ◽  
Reaction Forces

SUMMARY In three species of small therian mammals (Scandentia: Tupaia glis, Rodentia: Galea musteloides and Lagomorpha: Ochotona rufescens) the net joint forces and torques acting during stance phase in the four kinematically relevant joints of the forelimbs (scapular pivot,shoulder joint, elbow joint, wrist joint) and the hindlimbs (hip joint, knee joint, ankle joint, intratarsal joint) were determined by inverse dynamic analysis. Kinematics were measured by cineradiography (150 frames s-1). Synchronously ground reaction forces were acquired by forceplates. Morphometry of the extremities was performed by a scanning method using structured illumination. The vector sum of ground reaction forces and weight accounts for most of the joint force vector. Inertial effects can be neglected since errors of net joint forces amount at most to 10 %. The general time course of joint torques is comparable for all species in all joints of the forelimb and in the ankle joint. Torques in the intratarsal joints differ between tailed and tail-less species. The torque patterns in the knee and hip joint are unique to each species. For the first time torque patterns are described completely for the forelimb including the scapula as the dominant propulsive segment. The results are compared with the few torque data available for various joints of cats(Felis catus), dogs (Canis lupus f. familiaris),goats (Capra sp.) and horses (Equus przewalskii f. caballus).

Improving hydrological forecasts through temporal hierarchal reconciliation

2021 ◽  
Author(s):  
Mohammad Sina Jahangir ◽  
John Quilty
Keyword(s):  
Least Squares ◽  
Performance Metrics ◽  
Moving Average ◽  
Weighted Least Squares ◽  
Absolute Error ◽  
Efficiency Index ◽  
Ordinary Least Squares ◽  
Data Driven ◽  
Wide Range ◽  
Different Levels

&lt;p&gt;Hydrological forecasts at different horizons are often made using different models. These forecasts are usually temporally inconsistent (e.g., monthly forecasts may not sum to yearly forecasts), which may lead to misaligned or conflicting decisions. Temporal hierarchal reconciliation (or simply, hierarchical reconciliation) methods can be used for obtaining consistent forecasts at different horizons. However, their effectiveness in the field of hydrology has not yet been investigated. Thus, this research assesses hierarchal reconciliation for precipitation forecasting due to its high importance in hydrological applications (e.g., reservoir operations, irrigation, drought and flood forecasting). Original precipitation forecasts (ORF) were produced using three different models, including &amp;#8216;automatic&amp;#8217; Exponential Time-Series Smoothing (ETS), Artificial Neural Networks (ANN), and Seasonal Auto-Regressive Integrated Moving Average (SARIMA). The forecasts were produced at six timescales, namely, monthly, 2-monthly, quarterly, 4-monthly, bi-annual, and annual, for 84 basins selected from the Canadian model parameter experiment (CANOPEX) dataset. Hierarchical reconciliation methods including Hierarchical Least Squares (HLS), Weighted Least Squares (WLS), and Ordinary Least Squares (OLS) along with the Bottom-Up (BU) method were applied to obtain consistent forecasts at all timescales.&lt;/p&gt;&lt;p&gt;Generally, ETS and ANN showed the best and worst performance, respectively, according to a wide range of performance metrics (root mean square error (RMSE), normalized RMSE (nRMSE), mean absolute error (MAE), normalized MAE (nMAE), and Nash-Sutcliffe Efficiency index (NSE)). The results indicated that hierarchal reconciliation has a dissimilar impact on the ORFs&amp;#8217; accuracy in different basins and timescales, improving the RMSE in some cases while decreasing it in others. Also, it was highlighted that for different forecast models, hierarchical reconciliation methods showed different levels of performance. According to the RMSE and MAE, the BU method outperformed the hierarchical methods for ETS forecasts, while for ANN and SARIMA forecasts, HLS and OLS improved the forecasts more substantially, respectively. The sensitivity of ORF to hierarchical reconciliation was assessed using the RMSE. It was shown that both accurate and inaccurate ORF could be improved through hierarchical reconciliation; in particular, the effectiveness of hierarchical reconciliation appears to be more dependent on the ORF accuracy than it is on the type of hierarchical reconciliation method.&lt;/p&gt;&lt;p&gt;While in the present work, the effectiveness of hierarchical reconciliation for hydrological forecasting was assessed via data-driven models, the methodology can easily be extended to process-based or hybrid (process-based data-driven) models. Further, since hydrological forecasts at different timescales may have different levels of importance to water resources managers and/or policymakers, hierarchical reconciliation can be used to weight the different timescales according to the user&amp;#8217;s preference/desired goals.&lt;/p&gt;

scholarly journals Optimality of the maximum likelihood estimator in astrometry

2018 ◽  
Vol 616 ◽  
pp. A95 ◽  
Author(s):  
Sebastian Espinosa ◽  
Jorge F. Silva ◽  
Rene A. Mendez ◽  
Rodrigo Lobos ◽  
Marcos Orchard
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Weighted Least Squares ◽  
Least Square ◽  
Performance Limits ◽  
Likelihood Estimator ◽  
Weighted Least Square ◽  
Wide Range ◽  
Close Form

Context. Astrometry relies on the precise measurement of the positions and motions of celestial objects. Driven by the ever-increasing accuracy of astrometric measurements, it is important to critically assess the maximum precision that could be achieved with these observations. Aims. The problem of astrometry is revisited from the perspective of analyzing the attainability of well-known performance limits (the Cramér–Rao bound) for the estimation of the relative position of light-emitting (usually point-like) sources on a charge-coupled device (CCD)-like detector using commonly adopted estimators such as the weighted least squares and the maximum likelihood. Methods. Novel technical results are presented to determine the performance of an estimator that corresponds to the solution of an optimization problem in the context of astrometry. Using these results we are able to place stringent bounds on the bias and the variance of the estimators in close form as a function of the data. We confirm these results through comparisons to numerical simulations under a broad range of realistic observing conditions. Results. The maximum likelihood and the weighted least square estimators are analyzed. We confirm the sub-optimality of the weighted least squares scheme from medium to high signal-to-noise found in an earlier study for the (unweighted) least squares method. We find that the maximum likelihood estimator achieves optimal performance limits across a wide range of relevant observational conditions. Furthermore, from our results, we provide concrete insights for adopting an adaptive weighted least square estimator that can be regarded as a computationally efficient alternative to the optimal maximum likelihood solution. Conclusions. We provide, for the first time, close-form analytical expressions that bound the bias and the variance of the weighted least square and maximum likelihood implicit estimators for astrometry using a Poisson-driven detector. These expressions can be used to formally assess the precision attainable by these estimators in comparison with the minimum variance bound.

Women Ankle Joint Torques Comparison between Bare Foot and High Heel

2013 ◽  
Vol 378 ◽  
pp. 382-386
Author(s):  
Hai Bin Liu ◽  
Zhi Qiang He ◽  
Wen Xue Yuan ◽  
Zhao Li Meng
Keyword(s):  
Ankle Joint ◽  
Muscle Force ◽  
Inverse Dynamics ◽  
Frontal Plane ◽  
Joint Torque ◽  
Force System ◽  
Joint Torques ◽  
High Heel ◽  
Internal Phase ◽  
The Stability

Objective: Research on ankle joint torques of healthy women with high heel compared with bare foot based on Inverse Dynamics. Methods: 12 women were recruited and tested by motion and force system. Kinematical, kinetic and personal segment parameter data were used to compute ankle joint torques and compare the differences between bare foot and high heel.Conclusion: compared with bare foot, It can infer that Soleus and Gastrocnemius access the contraction in advance and keep higher muscle force. Tibia Anterior and Posterior must have to make powerful contraction that could keep the ankle joint with higher torque. Compared with sagital and frontal plane, high heel doesnt change the joint torque in horizontal plane during the whole internal phase, but the fluctuations of torque value may influence the stability during normal level walking.

Improving Net Joint Torque Calculations Through a Two-Step Optimization Method for Estimating Body Segment Parameters

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Raziel Riemer ◽  
Elizabeth T. Hsiao-Wecksler
Keyword(s):  
Inverse Dynamics ◽  
Optimization Problems ◽  
Joint Torque ◽  
Optimization Approach ◽  
Regression Equations ◽  
Random Errors ◽  
Body Segment ◽  
Joint Torques ◽  
Body Segment Parameters ◽  
True Values

Two main sources of error in inverse dynamics based calculations of net joint torques are inaccuracies in segmental motions and estimates of anthropometric body segment parameters (BSPs). Methods for estimating BSP (i.e., segmental moment of inertia, mass, and center of mass location) have been previously proposed; however, these methods are limited due to low accuracies, cumbersome use, need for expensive medical equipment, and∕or sensitivity of performance. This paper proposes a method for improving the accuracy of calculated net joint torques by optimizing for subject-specific BSP in the presence of characteristic and random errors in motion data measurements. A two-step optimization approach based on solving constrained nonlinear optimization problems was used. This approach minimized the differences between known ground reaction forces (GRFs), such as those measured by a force plate, and the GRF calculated via a top-down inverse dynamics approach. In step 1, a series of short calibration motions was used to compute first approximations of optimized segment motions and BSP for each motion. In step 2, refined optimal BSPs were derived from a combination of these motion profiles. We assessed the efficacy of this approach using a set of reference motions in which the true values for the BSP, segment motion, GRF, and net joint torques were known. To imitate real-world data, we introduced various noise conditions on the true motion and BSP data. We compared the root mean squared errors in calculated net joint torques relative to the true values due to the optimal BSP versus traditionally-derived BSP (from anthropometric tables derived from regression equations) and found that the optimized BSP reduced the error by 77%. These results suggest that errors in calculated net joint torques due to traditionally-derived BSP estimates could be reduced substantially using this optimization approach.

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