asimpledog参数估计结果

morphology pose

8 morphology poses

Poses	Meaning
circular_000	Hip rotation at approximately 0°
circular_030	Mirrored hip rotations at ±30°
circular_060	Mirrored hip rotations at ±60°
circular_090	Mirrored hip rotations at ±90°
knees_axial_outward	Knees pointing outward along the body axis
knees_axial_inward	Knees pointing inward along the body axis
knees_radial_outward	Knees pointing radially away from body center
knees_radial_inward	Knees pointing radially toward body center

Stiffness&Damping

Static Stiffness

K_{p, i}^{static} = \frac{| τ_{hold, i} |}{Δ q_{allow, i}}

Static Damping

Each joint is independently approximated as a second-order system:

J_{eff} \ddot{q} + K_{d} \dot{q} + K_{p} q = 0

Undamped natural angular frequency:

ω_{n} = \sqrt{\frac{K_{p}}{J_{eff}}}

Undamped natural frequency in hertz:

f_{n} = \frac{ω_{n}}{2 π}

Critical damping coefficient:

K_{d, critical} = 2 \sqrt{K_{p} J_{eff}}

Damping ratio:

ζ = \frac{K_{d}}{2 \sqrt{K_{p} J_{eff}}}

Given $K_{p}$ and the desired damping ratio, $K_{d}$ is calculated as:

K_{d, i} = 2 ζ_{i} \sqrt{K_{p, i} J_{eff, i}}

where $J_{eff}$ denotes the effective joint inertia:

$J_{locked, j} = M_{j j}$ : joint inertia when the base and all other generalized coordinates are locked.
$J_{free, j} = \frac{1}{(M^{- 1})_{j j}}$ : effective joint inertia when the base and all other joints are free. The other generalized coordinates are unforced and allowed to move through dynamic coupling.

Calculation：

q_allow = 5° = 0.0872664626 rad
zeta = 1.0
J_eff = locked

Joint Family	$K_{p}$ Range	$K_{d}$ Range
hip_flexion	0.428391–53.2874	0.276606–3.70349
hip_abduction	0.0329987–50.4153	0.0803375–3.38429
hip_rotation	0.278956–37.1817	0.104658–1.20771
knee_flexion	24.7168–102.117	0.959010–1.94929

Selected upper-bound gains:

Hip: $K_{p} = 50$ , $K_{d} = 3.38$
Knee: $K_{p} = 100$ , $K_{d} = 1.95$

IsaacSim DCMotor

DC Motor in Isaac Lab

#	Parameter	Description	Unit	$K_{p} = 50$ Configuration	$K_{p} = 100$ Configuration
1	`stiffness`	Position-control gain $K_{p}$	N·m/rad	50.0	100.0
2	`damping`	Velocity-feedback damping gain $K_{d}$	N·m/(rad/s)	3.38	1.95
3	`armature`	Rotor inertia reflected to the output shaft	kg·m²	0.0012484886	0.0012484886
4	`friction`	Static/breakaway friction torque	N·m	0.2384994984	0.1074789484
5	`dynamic_friction`	Coulomb friction torque during motion	N·m	0.00126407998	0.000537669324
6	`viscous_friction`	Viscous friction coefficient	N·m/(rad/s)	2.3799067e-5	1.0411770e-5
7	`effort_limit`	Continuous output torque limit	N·m	36.0	36.0
8	`saturation_effort`	Stall/saturation torque	N·m	72.0	72.0
9	`velocity_limit`	No-load speed limit	rad/s	53.0	53.0

Torque control:

τ_{computed} = K_{p} (q_{des} - q) + K_{d} ({\dot{q}}_{des} - \dot{q}) + τ_{ff}

Effort/velocity

For positive torque, the torque–speed envelope is defined by:

τ_{max} (\dot{q}) = min [τ_{effort}, τ_{sat} (1 - \frac{| \dot{q} |}{{\dot{q}}_{max}})]

Based on the RS 06 torque–speed curve, the desired characteristics are:

A peak torque of approximately 36 N·m at low speeds.
Torque remaining near its peak up to approximately 250 rpm.
Torque decreasing to nearly zero at approximately 510 rpm.

Accordingly, the following parameter values can be used:

effort_limit=36.0, 
saturation_effort=72.0, 
velocity_limit=53.0,    # rad/s = RPM * 2 * pi / 60

Motor rotor armature

\begin{aligned} m_{r} & = ρ π h (r_{o}^{2} - r_{i}^{2}) \\ J_{r} & = \frac{1}{2} m_{r} (r_{o}^{2} + r_{i}^{2}) \\ J_{out} & = N^{2} J_{r} \end{aligned}

Axial length: h=25 mm
Outer radius: r_o=15 mm
Inner radius: r_i=5 mm
Effective density: ρ=7850 kg/m³
Gear reduction ratio: N=9

Calculate：

\begin{aligned} m & = ρ π h (r_{o}^{2} - r_{i}^{2}) \approx 0.12331 kg \\ J_{rotor} & = \frac{1}{2} m (r_{o}^{2} + r_{i}^{2}) \approx 1.54134 \times 10^{- 5} kg m^{2} \end{aligned}

J_{out} = N^{2} J_{rotor} = 9^{2} \times 1.54134 \times 10^{- 5} \approx 1.2484886 \times 10^{- 3} kg m^{2}

  "assumptions": {
    "axial_length_m": 0.025,
    "outer_radius_m": 0.015,
    "inner_radius_m": 0.005,
    "equivalent_density_kg_m3": 7850.0,
    "gear_ratio_motor_speed_per_output_speed": 9.0
  },
  "calculation": {
    "mass_expression": "rho*pi*h*(r_outer^2-r_inner^2)",
    "rotor_inertia_expression": "0.5*mass*(r_outer^2+r_inner^2)",
    "output_side_expression": "rotor_inertia*gear_ratio^2",
    "equivalent_mass_kg": 0.12330751165392173,
    "rotor_inertia_kg_m2": 1.541343895674022e-05,
    "output_side_armature_kg_m2": 0.001248488555495958
  },

Motor parameter identification

Payload model

两个模型的底座相同：

底座质量：0.5282673472 kg
底座质心：[0.0002567991, -0.0000035019, 0.0202477424] m
底座惯量：Ixx=0.0004119872、Iyy=0.0004178833、Izz=0.0005423930 kg·m²

2.1 负载平衡模型

Payload 质量：2.4072134493 kg
质心：[-4.14334e-6, -1.51857e-5, -0.04553585] m
Ixx = 0.05500499938 kg·m²
Iyy = 0.003552246411 kg·m²
Izz = 0.05767903458 kg·m²
Ixy = 1.54564e-6 kg·m²
Ixz = 1.93840e-7 kg·m²
Iyz = 7.59740e-7 kg·m²
按平行轴定理换算到关节 Z 轴：约 0.05767903518 kg·m²
q=0 附近重力矩：约 0.0003586 N·m，基本平衡

2.2 单边负载模型

Payload 质量：1.4082332749 kg
质心：[9.95192e-6, -0.1064336763, -0.0424325965] m
质心到关节轴的径向距离：约 0.106434 m
Ixx = 0.01485312262 kg·m²
Iyy = 0.001884312992 kg·m²
Izz = 0.01611149247 kg·m²
Ixy = -3.54392e-6 kg·m²
Ixz = -2.59302e-7 kg·m²
Iyz = -0.001119882209 kg·m²
按平行轴定理换算到关节 Z 轴：约 0.03206413861 kg·m²
q=0 最大重力矩：约 1.4703566 N·m

Optimization

Optimized parameters: friction, dynamic_friction, and viscous_friction.

Parameter bounds:

friction: $[0.02, 0.45]$ N·m
Dynamic-to-static friction ratio: $[0.005, 0.60]$
viscous_friction: $[1 \times 10^{- 5}, 0.05]$ N·m/(rad/s)

Log-space normalization:

\begin{aligned} u (x) & = \frac{\ln x - \ln x_{min}}{\ln x_{max} - \ln x_{min}}, \\ x (u) & = \exp [\ln x_{min} + u (\ln x_{max} - \ln x_{min})], \\ F_{d} & = F_{s} ρ, ρ = \frac{F_{d}}{F_{s}} . \end{aligned}

Objective Function

The position root-mean-square error (RMSE) for each trajectory is defined as:

R_{i} = \sqrt{\frac{1}{T_{i}} \sum_{t = 1}^{T_{i}} {(q_{i, t}^{sim} - q_{i, t}^{real})}^{2}}

The Huber loss function is defined as:

h_{δ} (z) = {\begin{cases} \frac{1}{2} z^{2}, & | z | \leq δ, \\ δ (| z | - \frac{1}{2} δ), & | z | > δ, \end{cases} δ = 1.5

The velocity-error term is defined as:

H_{v, i} = \frac{1}{T_{i}} \sum_{t = 1}^{T_{i}} h_{1.5} (\frac{v_{i, t}^{sim} - v_{i, t}^{real}}{σ_{v}})

The final objective function is:

\begin{aligned} L (θ) = & 0.70 {mean}_{i = 1}^{12} (\frac{R_{i}}{0.02}) \\ + 0.30 mean ({Top 3}_{i = 1}^{12} \frac{R_{i}}{0.02}) \\ + 0.01 {mean}_{i = 1}^{12} (H_{v, i}) \end{aligned}

Data

ID	Trajectory Name	Trajectory Profile	Primary Purpose	Use in the Current Standard Workflow
R01	`balance_holds`	Holds at $- 0.2$ , $0$ , and $+ 0.2$ rad	Evaluate load balancing, static bias, holding error, and breakaway/static friction	Fit
R02	`micro_sines`	Low-amplitude sinusoids of 0.01/0.03/0.06 rad at 0.1/0.2/0.4 Hz	Excite breakaway friction, dead zones, small-signal tracking, and stick–slip behavior	Fit
R03	`multiscale_smooth_steps`	Multiscale smooth steps of 0.25/0.5/1.0 rad	Evaluate control latency, effective $K_{p} / K_{d}$ , rise time, overshoot, and transient response	Fit
R04	`rounded_triangle`	Rounded triangular wave, ±0.2 rad at 0.05 Hz	Excite low-speed friction, direction reversals, hysteresis, and backlash	Fit
R05	`wide_constant_velocity_round_trips`	Bidirectional constant-velocity sweeps at 0.1–4.0 rad/s	Distinguish static friction, dynamic Coulomb friction, and velocity-dependent resistance	Fit
R06	`wide_minimum_jerk_acceleration_sweeps`	Large-amplitude minimum-jerk motions of ±0.8 rad with varying durations	Excite `armature` inertia, load inertia, and acceleration dynamics	Fit
R07	`variable_amplitude_log_chirp_up`	Upward logarithmic chirp from 0.1 to 3.0 Hz with frequency-dependent amplitude	Characterize broadband closed-loop dynamics and magnitude/phase responses	Fit
R08	`variable_amplitude_log_chirp_down`	Downward logarithmic chirp from 3.0 to 0.1 Hz	Evaluate sweep-direction dependence, thermal drift, hysteresis, and time-varying effects	Fit
R09	`multisine_small`	Low-amplitude multisine excitation from 0.1 to 2.5 Hz, with a peak amplitude of approximately 0.15 rad	Provide small-to-moderate-amplitude broadband periodic excitation and evaluate the linear closed-loop response	Fit
R10	`multisine_large`	Same frequencies and phases as R09, with a peak amplitude of approximately 0.45 rad	Compare with R09 to identify amplitude-dependent nonlinearities	Fit
R11	`wide_filtered_binary_sequence`	Smoothed random binary switching with an amplitude of ±0.3 rad	Excite broadband dynamics, control latency, random direction reversals, and aperiodic residuals	Fit
R12	`held_out_sine_bank`	Multiple independent sinusoidal segments from 0.15 to 2.9 Hz, with amplitudes decreasing from 0.8 to 0.12 rad	Evaluate frequency points and large-amplitude responses not directly covered by the other trajectories	Promoted to Fit in the current standard workflow
R13	`smooth_clipped_sine`	Smoothly clipped 1 Hz sinusoid	Evaluate clipping nonlinearities, transition dynamics, and overall system generalization	Sole held-out validation trajectory

Trajectory	Name	Sample Count	Duration
R01	`balance_holds`	1,400	28.00 s
R02	`micro_sines`	3,075	61.50 s
R03	`multiscale_smooth_steps`	1,800	36.00 s
R04	`rounded_triangle`	2,000	40.00 s
R05	`wide_constant_velocity_round_trips`	6,103	122.06 s
R06	`wide_minimum_jerk_acceleration_sweeps`	1,372	27.44 s
R07	`variable_amplitude_log_chirp_up`	2,250	45.00 s
R08	`variable_amplitude_log_chirp_down`	2,250	45.00 s
R09	`multisine_small`	2,500	50.00 s
R10	`multisine_large`	2,500	50.00 s
R11	`wide_filtered_binary_sequence`	2,000	40.00 s
R12	`held_out_sine_bank`	4,556	91.12 s
R13	`smooth_clipped_sine`	1,250	25.00 s

数据范围	单个 Kp 条件样本数	单个 Kp 条件时长
R01–R11	27,250	545.00 s（9分05秒）
R12–R13	5,806	116.12 s（1分56.12秒）
R01–R12，当前 CMA fit	31,806	636.12 s（10分36.12秒）
R13，held-out validation	1,250	25.00 s
R01–R13 总计	33,056	661.12 s（11分01.12秒）

Result

Parameter	$K_{p} = 50$ , $K_{d} = 3.38$	$K_{p} = 100$ , $K_{d} = 1.95$
`armature` (fixed)	0.0012484886	0.0012484886
`friction` [N·m]	0.2384994984	0.1074789484
`dynamic_friction` [N·m]	0.00126407998	0.000537669324
Dynamic-to-static friction ratio	0.00530014	0.00500255
`viscous_friction` [N·m/(rad/s)]	2.3799067e-5	1.0411770e-5
Optimal objective value	0.848628998	1.002159834
Median fit RMSE [rad]	0.00894282	0.01402229
90th-percentile fit RMSE [rad]	0.02249461	0.03209888
Fits with RMSE $\leq 0.02$ rad	9/12	9/12
R13 RMSE [rad]	0.02581465	0.03185880

Final parameters

#	Parameter	Description	Unit	$K_{p} = 50$ Configuration	$K_{p} = 100$ Configuration
1	`stiffness`	Position-control gain $K_{p}$	N·m/rad	50.0	100.0
2	`damping`	Velocity-feedback damping gain $K_{d}$	N·m/(rad/s)	3.38	1.95
3	`armature`	Rotor inertia reflected to the output shaft	kg·m²	0.0012484886	0.0012484886
4	`friction`	Static/breakaway friction torque	N·m	0.2384994984	0.1074789484
5	`dynamic_friction`	Coulomb friction torque during motion	N·m	0.00126407998	0.000537669324
6	`viscous_friction`	Viscous friction coefficient	N·m/(rad/s)	2.3799067e-5	1.0411770e-5
7	`effort_limit`	Continuous output torque limit	N·m	36.0	36.0
8	`saturation_effort`	Stall/saturation torque	N·m	72.0	72.0
9	`velocity_limit`	No-load speed limit	rad/s	53.0	53.0

DCMotorCfg(
    joint_names_expr=["hip"],
    stiffness=50.0,
    damping=3.38,
    armature=0.0012484886,
    friction=0.238499498429,
    dynamic_friction=0.00126407997987,
    viscous_friction=2.37990671357e-05,
    effort_limit=36.0,
    saturation_effort=72.0,
    velocity_limit=53.0,
)

DCMotorCfg(
    joint_names_expr=["knee"],
    stiffness=100.0,
    damping=1.95,
    armature=0.0012484886,
    friction=0.107478948424,
    dynamic_friction=0.000537669323874,
    viscous_friction=1.04117699332e-05,
    effort_limit=36.0,
    saturation_effort=72.0,
    velocity_limit=53.0,
)

Something interesting

低速的周期阻滞能识别出来
可能不只是滑动摩擦在影响电机表现：Translational Friction | AnyMath 文档还可能有低速下的Shtribek摩擦