Loading the robot…加载机器人模型…
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My quadruped walks on four printed lower legs, each turned by an SG90 knee servo. Inverse design with deep learning: a conditional neural implicit field turns performance targets into a printable leg in 2–5 seconds. It learned from 3D topology optimisation: 80 legs, each scored by finite elements.我的四足机器人靠 4 条打印小腿行走，每条由膝关节的 SG90 舵机驱动。用深度学习做逆向设计：条件神经隐式场由性能目标在 2–5 秒内生成可打印的腿。训练数据来自 3D 拓扑优化：80 条小腿，每条都经有限元评分。
Result · v8最终结果 · v8
Generated by the deep-learning inverse model, trained on 80 topology-optimised legs, from four targets: stance and push stiffness no lower than the original’s, knee torque no higher, leg length 47.1 ± 1 mm, least mass. Checked as solid models by finite elements and the gait’s kinematics.由深度学习逆向模型按目标生成（以 80 条拓扑优化结果训练）：站立与推地刚度不低于原版，膝关节扭矩不高于原版，腿长 47.1 ± 1 mm，质量最小。按实心模型，用有限元和步态运动学复核。
The quadruped has eight SG90 servos: four at the hips, four at the knees. Each knee servo turns one printed lower leg, R3, whose end meets the ground. This project redesigns that lower leg twice over: forward, by topology optimisation; inverse, by deep learning. The rest of the robot stays as it is.这台四足机器人有 8 个 SG90 舵机：4 个在髋关节，4 个在膝关节。每个膝关节舵机带动一条打印小腿 R3，小腿末端触地。本项目用两种方法重新设计这条小腿：正向用拓扑优化，逆向用深度学习；机器人其余部分不变。
Hip · SG90髋关节 · SG90Turns the upper leg带动大腿
Knee · SG90膝关节 · SG90Plastic gears; turns the lower leg塑料齿轮，带动小腿
Lower leg R3 · redesigned小腿 R3 · 优化对象Original: 5.84 g solid原版实心 5.84 g
Contact point触地点Grip depends on the sole抓地取决于脚底
Controller主控板ESP32-S3, servo driverESP32-S3 与舵机驱动
Four identical lower legs4 条小腿相同One part on every leg每条腿同一零件
Hip · SG90髋关节 · SG90Turns the upper leg带动大腿
Knee · SG90膝关节 · SG90Plastic gears; turns the lower leg塑料齿轮，带动小腿
Lower leg R3 · redesigned小腿 R3 · 优化对象Original: 5.84 g solid原版实心 5.84 g
Contact point触地点Grip depends on the sole抓地取决于脚底
Controller主控板ESP32-S3, servo driverESP32-S3 与舵机驱动
Four identical lower legs4 条小腿相同One part on every leg每条腿同一零件
The knee SG90 has plastic gears. The lighter the leg and the closer its mass sits to the knee, the less torque the gears carry. Measures: mass and knee inertia ↓.膝关节 SG90 是塑料齿轮。小腿越轻、质量离膝越近，齿轮承受的力矩越小。指标：质量、绕膝转动惯量 ↓。
Upgrades will add parts, and the legs will carry more. The first round holds the original’s stiffness as a floor at the least mass; for more, the inverse model generates to the new targets: a leg with at least 1.95× stance and 1.80× push stiffness weighs 5.67 g, still lighter than the original.升级会加装更多部件，腿的载荷随之增加。第一轮以原版刚度为下限、质量最小；需要更高刚度时，逆向模型按新目标生成：站立 ≥ 1.95 倍、推地 ≥ 1.80 倍的腿为 5.67 g，仍比原版轻。
Grip is set by the sole’s shape and the floor, and finite elements cannot compute friction. The sole radius is a design condition; grip is measured in an A/B walk against the original on a wood floor, and the result becomes a target of the second round.抓地由脚底形状和地面决定，有限元无法计算摩擦。脚底半径是设计条件之一；抓地在木地板上与原版做 A/B 行走实测，实测结果作为第二轮的目标。
Constraints: knee torque no higher than the original’s, and a leg length of 47.1 ± 1 mm, so the firmware gait stays as it is.约束：膝关节扭矩不高于原版，腿长 47.1 ± 1 mm，固件步态无需重调。
Forward design正向设计
The interfaces and keep-outs are fixed; the optimiser distributes the material; every result is scored the same way as the original; the run is repeated across the condition space.接口和禁区固定，材料分布由优化器决定；每个结果与原版采用同一评分；在整个条件空间内重复。
Design domain: where material may go, inside a 40° ground cone设计域：40° 着地锥内可布置材料的区域
Keep-out: the SG90 body (12.5 × 23 mm) and its cable gap禁区：SG90 机身（12.5 × 23 mm）及走线缺口
Keep-in: 1.6 mm grip walls, 4 mm tab seats, the sole; the servo presses in, no screws保留区：1.6 mm 夹持壁、4 mm 耳座、脚底；舵机压入固定，无螺丝
The upper leg’s sweep over ±60° of knee travel, plus 1 mm大腿件在膝关节 ±60° 内的扫掠区，外扩 1 mm
Knee axis (the servo shaft), 47.1 mm above the contact point膝关节轴（舵机输出轴），距着地点 47.1 mm
Stance cases: 4 N up through the foot, leg at −20°, 0°, 20°, 40°站立工况：脚底向上 4 N，腿倾角 −20°、0°、20°、40°
Push case: 6 N across the 9.5 mm thickness推地工况：6 N，沿 9.5 mm 厚度方向
The load-bearing shank is the top third of the original part; the holder is the same in every design.承力的小腿杆只占原版零件的上三分之一；舵机架在所有设计中保持不变。
The hip swings within 8° of the knee axis, so the push loads the plate in its weakest direction.髋关节摆动方向与膝关节轴仅差 8°，推地力作用在平板最弱的方向。
Integrated over the part’s 0.4 mm voxels, PLA at 1.24 g/cm³; the inertia is taken about the knee axis.在零件的 0.4 mm 体素上积分，PLA 密度 1.24 g/cm³；转动惯量绕膝关节轴计算。
Four stance cases and the push, on the same grid as the original. Every leg has a safety factor of at least 12 where 3 is required, so strength never binds.四个站立工况和推地工况，与原版在同一网格上计算。所有腿安全系数 ≥ 12（要求 3），强度不构成限制。
The firmware’s six-frame gait on the robot’s kinematic chain; all three depend only on leg length and sole radius.由固件六帧步态在机器人运动链上计算；三者只取决于腿长和脚底半径。
On a wood floor against the original legs: forward travel, sideways drift and turning per cycle.木地板上与原版腿对比：每个周期的前进、横移与转角。
Every leg, the original included, is scored as a solid model, so print settings apply to all alike. The part is voxelised at 0.4 mm and solved with eight-node bricks carrying Wilson’s incompatible modes (PLA, E = 2750 MPa, ν = 0.35), fixed where the servo touches it.所有腿（含原版）按实心模型评分，打印参数对所有腿相同。零件以 0.4 mm 体素化，采用带 Wilson 非协调模式的八节点六面体单元求解（PLA，E = 2750 MPa，ν = 0.35），舵机接触面固定。
Stiffness is the load over the mean displacement along it. The stance score is the mean of the four stance ratios to the original; the push score is the push ratio. Strength is 35 MPa over the 99th-percentile von Mises stress across all cases.刚度 = 载荷 ÷ 载荷方向的平均位移。站立分数为四个站立工况相对原版比值的平均，推地分数为推地工况的比值。强度 = 35 MPa ÷ 全部工况第 99 百分位 von Mises 应力。
A 19-parameter family of branching members, 1,440 designs searched with NSGA-II: 2.3× the original’s stiffness at its mass, or 30 % lighter at its stiffness.19 参数的分叉杆件族，NSGA-II 评估 1,440 个设计：同质量下刚度为原版的 2.3 倍，同刚度下轻 30%。
A density per 0.6 mm cell of the side profile, extruded through the thickness. The optimiser again draws two rails; its front lies on the parametric one.侧面轮廓按 0.6 mm 网格分配密度，沿厚度拉伸。优化器同样得到两根杆件，前沿与参数化结果重合。
With the thickness free, 23 of the 79 legs close part of the shank into a ring, which resists bending and torsion in every direction, the push included.厚度方向放开后，79 条腿中有 23 条的小腿杆局部闭合为环形截面，可同时抵抗各方向的弯曲和扭转，包括推地方向。
0.8 mm cells, 0.79 mm through the thickness: the 9.5 mm plate in 12 layers plus 4 each side; 48 × 79 × 20 in all.0.8 mm 网格，厚度方向 0.79 mm：9.5 mm 板厚 12 层，两侧各加 4 层；共 48 × 79 × 20。
Each case’s compliance over the original’s; push weight w, each stance case (1 − w) / 4.各工况柔度除以原版柔度；推地权重 w，四个站立工况各 (1 − w) / 4。
The budget sets the material fraction; keep-in cells are fixed solid, keep-out cells empty.由质量预算换算体积分数；保留区固定为实，禁区固定为空。
Density filter of 2.4 mm radius, empty outside the domain; tanh projection, β doubled every 10 iterations from the 20th, 1 → 16.密度滤波半径 2.4 mm，设计域外视为空；tanh 投影，β 自第 20 步起每 10 步加倍，1 → 16。
Penalty 3, minimum stiffness 10⁻³, move limit 0.08, 80 iterations; one factorisation solves all five cases.惩罚指数 3，最小刚度 10⁻³，步长 0.08，80 步；一次分解求解五个工况。
About 10 minutes on four cores with PARDISO; 30–50 on a MacBook Air M2 with algebraic multigrid.4 核 PARDISO 约 10 分钟；MacBook Air M2 代数多重网格 30–50 分钟。
Resampled to 0.4 mm, blurred by 0.6 mm, thresholded at constant volume重采样至 0.4 mm，0.6 mm 高斯模糊，按体积守恒取阈值
Smoothed by 0.5 mm; a 3 mm smooth union joins the sole0.5 mm 平滑；3 mm 平滑并集连接脚底
Marching cubes at 0.4 mm, 5 rounds of Taubin smoothing0.4 mm 移动立方体，Taubin 平滑 5 轮
Servo holder joined, pocket and flanges cut, flat at the ground (manifold3d)并入舵机架，切出窗口与法兰位，着地面切平（manifold3d）
The finished part is one watertight body, re-voxelised and scored at 0.4 mm; the dataset records the finished part’s scores.成品为单一水密实体，重新体素化后以 0.4 mm 网格评分；数据集记录的是成品的分数。
Interactive可交互
The same optimiser in 2D: plane-stress elements, the four stance cases, the same filter, projection and update. Set the mass budget, leg length and sole radius.同一优化器的 2D 版本：平面应力单元、四个站立工况，滤波、投影和更新规则相同。可调整质量预算、腿长和脚底半径。
Loading the optimiser…加载优化器…
Four conditions span the space: mass budget 3.5–6.5 g, leg length 40–54 mm, push weight 0.2–0.8, sole radius 6–60 mm (log scale). 64 designs follow a Sobol sequence; 16 more sample the region of the inverse question (3.5–4.8 g, 46.1–48.1 mm).4 个条件构成设计空间：质量预算 3.5–6.5 g，腿长 40–54 mm，推地权重 0.2–0.8，脚底半径 6–60 mm（对数尺度）。64 个设计按 Sobol 序列采样，另 16 个加密在逆向问题所在区域（3.5–4.8 g，46.1–48.1 mm）。
79 are valid; in one, the sole cannot connect to the servo holder. At the original’s mass, the stiffest design (s043) is 4.5× as stiff in stance and 6.9× in push.79 个有效；1 个脚底无法与舵机架连通。与原版同质量时，最刚的设计（s043）站立刚度为 4.5 倍，推地为 6.9 倍。
Stance stiffness ÷ original站立刚度 ÷ 原版
Push stiffness ÷ original推地刚度 ÷ 原版
Solid mass, g实心质量，g
Eight Gaussian processes (squared-exponential kernel, one length scale per condition) predict every measure from the four conditions. Leave-one-out error as a share of each measure’s range: torque 2.5 %, stride 2.5 %, lift 0.5 %, stance 7.3 %, push 8.7 %, inertia 9.2 %, mass 13.5 %.8 个高斯过程（平方指数核，每个条件独立长度尺度）由 4 个条件预测各项性能。留一法误差占各项分布范围：扭矩 2.5%，步长 2.5%，抬脚 0.5%，站立 7.3%，推地 8.7%，惯量 9.2%，质量 13.5%。
Interactive可交互
Set the four conditions to bring up the nearest optimised leg and its finite-element scores, or pick a point on the chart.调整 4 个条件，显示最接近的优化结果及其有限元分数；也可在图中直接选择。
Loading the 79 legs…加载 79 个优化结果…
Inverse design · deep learning逆向设计 · 深度学习
One optimisation answers one set of conditions in 10–50 minutes. The inverse model answers the reverse question, the lightest leg that meets given targets, in 2–5 seconds.一次优化对应一组条件，耗时 10–50 分钟。逆向模型求解相反的问题：给定目标，生成满足目标且质量最小的腿，耗时 2–5 秒。
The first round puts the servo first: the original’s stiffness as a floor, the least mass. Strength does not bind (the weakest leg has a safety factor of 12), and the stiffness margin for later payload is a new target for the same model.第一轮以舵机负载为先：刚度以原版为下限，质量最小。强度不构成限制（最弱的腿安全系数为 12）；为后续加装预留的刚度余量，作为新目标交给同一个模型。
The torque limit also sets the sole: at 47.1 mm, the knee torque stays at or below the original’s only for sole radii of about 22 mm or more (6 mm: 22.8 N·mm).扭矩约束同时限定了脚底：腿长 47.1 mm 时，脚底半径约 22 mm 以上，膝关节扭矩才不超过原版（6 mm 时为 22.8 N·mm）。
The eight Gaussian processes predict every measure, with its standard deviation σ, for 20,000 random candidates (sole radius sampled on a log scale).8 个高斯过程预测条件空间内 2 万个随机候选（脚底半径对数采样）的各项性能及其标准差 σ。
A candidate qualifies when μ − σ meets each minimum and μ + σ stays under each maximum; the lightest predicted mass wins.下限目标满足 μ − σ ≥ 目标、上限目标满足 μ + σ ≤ 目标即为合格；取预测质量最小者。
A conditional neural implicit field gives the signed distance to the surface at any point, for the chosen conditions (below).条件神经隐式场：给定条件下，输出任意点到表面的符号距离（见下）。
Decoded on 556,800 grid points and meshed; the surface is offset by bisection within ±0.4 mm until the mass matches the surrogate’s prediction within 0.5 %. Without this, light legs come out 5 % thin and 20 % soft.在 556,800 个网格点上解码并提取表面；表面在 ±0.4 mm 内二分偏移，直到质量与代理模型预测相差 < 0.5%。不做这一步时，轻腿偏薄 5%、刚度偏低 20%。
Finishing as in forward design: Taubin smoothing, exact servo interfaces.后处理与正向设计相同：Taubin 平滑，精确切出舵机接口。
Scored with the dataset’s 0.4 mm finite elements.以与数据集相同的 0.4 mm 有限元评分。
If a target is missed, the conditions are picked again at 1.75σ, then 2.5σ; up to three tries.未达标时以 1.75σ、再以 2.5σ 重选条件；最多 3 次。
The shape is a signed distance field f(p, c): p is a point in space, c the four conditions, and the output is p’s distance to the surface, negative inside. The surface is the level set f = 0. One network holds every design: shapes between the optimised conditions are its interpolation, and the decoding grid sets the resolution.形状用符号距离场 f(p, c) 表示：p 为空间中的点，c 为 4 个条件，输出 p 到表面的距离，内部为负；表面是 f = 0 的等值面。一个网络表示全部设计：优化样本之间的形状由网络插值得到，分辨率由解码网格决定。
Input: the point, its sines and cosines at six frequencies (2ᵏπ, k = 0…5) so the network can draw thin members, and the four scaled conditions: 43 numbers. Six layers of 256 with SiLU; the 43 inputs join again at the fourth layer. Output: the distance, clipped at ±4 mm. 351,489 trained weights.输入：点坐标、6 个频率（2ᵏπ，k = 0…5）的正余弦编码（使网络能表达细杆件）和 4 个归一化条件，共 43 维。6 层 × 256，SiLU 激活；43 维输入在第 4 层再次接入。输出：截断于 ±4 mm 的距离。共 351,489 个训练参数。
Interactive可交互
Pick a point on the section to see its activation in every layer and the distance it outputs; move the four conditions and the contour follows.在截面上选一个点，查看它在每一层的激活和输出的距离；调整 4 个条件，等值线随之变化。
Loading the network…加载神经网络…
Training loss (L1, weight 5 near the surface)训练损失（L1，表面附近权重 5）
Step步数
Interactive可交互
The same Gaussian processes and 20,000 candidates pick the conditions; the same network generates the shape on your GPU. The defaults are the first-round targets and return v8’s conditions.同一组高斯过程与 2 万个候选选出条件，同一网络在本机 GPU 上生成形状。默认值为第一轮目标，输出 v8 的条件。
Loading the model and the network…加载代理模型与神经网络…
s039 · IoU 0.94s039 · IoU 0.94
s024 · IoU 0.90s024 · IoU 0.90
R3R3
Optimised优化结果
Generated逆向生成
Optimised优化结果
Generated逆向生成
Original原版
Twenty target sets, spread evenly: stance 1–2×, push 1–2.5×, knee torque at most 8–16 N·mm, leg 46.1–48.1 mm. All twenty were met, seventeen at the first try. Every generated leg is lighter than the lightest dataset leg meeting the same targets (median −7 %, −1 % to −23 %): the model finds designs between the optimisation samples.20 组目标均匀分布：站立 1–2 倍，推地 1–2.5 倍，膝关节扭矩上限 8–16 N·mm，腿长 46.1–48.1 mm。20 组全部达标，17 组一次达标。每条生成的腿都比数据集中满足相同目标的最轻设计更轻（中位数 −7%，−1% 至 −23%）：模型生成的是优化样本之间的设计。
Loading the model…加载模型…
Conditions: mass budget 3.92 g, leg 46.3 mm, push weight 0.24, sole radius 21.2 mm. The stride matches the original’s (11.33 against 11.36 mm), so in the A/B walk the gaits differ only in foot lift, 1.4 mm lower on the rocker sole.条件：质量预算 3.92 g，腿长 46.3 mm，推地权重 0.24，脚底半径 21.2 mm。步长与原版一致（11.33 / 11.36 mm），A/B 中两组步态仅抬脚高度不同：圆弧脚底低 1.4 mm。
A knee-torque limit of 10 N·mm costs 0.07 g more and shortens the stride by 6.5 %; that leg is held for the second round, so the first A/B compares grip alone.将扭矩上限降至 10 N·mm 仅增加 0.07 g，但步长缩短 6.5%；该设计留待第二轮，使第一轮 A/B 只比较抓地。