The skeletal proportions we assumed for Caudipteryx were based primarily on BPM (Beipiao Paleontological Museum, Beipiao, China) 0001, a specimen with a complete, well-preserved skeleton but no associated plumage. The dimensions of the wing plumage were based primarily on measurements from IVPP (Institute of Vertebrate Paleontology and Paleoanthropology, Beijing, China) V12430 and V12344, specimens in which the forelimb feathers are relatively well preserved. The wing outline reconstructed in our study is shown in the inset to Fig. 1. We assumed the cross-sectional shape of the wing surface at various points between the shoulder joint and the wing tip to be similar to that seen in modern birds. The mediolateral length and the area of the reconstructed wing are respectively 0.24 m and 0.01797 m2, the aspect ratio of the fully unfolded wing is 0.32, and the average chord (i.e. anteroposterior width across the wing surface) of the wing is 0.10 m. Some parts of our analysis did not take the full three-dimensional structure of the reconstructed wing into account, but all parts were based on either the full reconstructed wing or some simplification of its geometry. In many cases aspect ratio was used as a proxy for the amount of wing unfolding, with larger aspect ratios corresponding to larger wing areas (because the wing was assumed to be more fully deployed).

We estimated the maximum running speed of Caudipteryx to be 8 m/s. This value was based on the skeletal hindlimb proportions of BPM 0001, and on adopting the assumptions used in Hutchinson (2004) with respect to the limb posture of small theropods and the range of Froude numbers (up to 17) they might have utilized in running. However, calculations in some parts of the analysis were carried out for running speeds of up to 10 m/s, in order to allow for the possibility that Caudipteryx could run somewhat faster than our method of estimation would indicate.

Classical aerodynamic analysis of the wing of Caudipteryx during terrestrial running

In the first phase of our theoretical analysis of the wings of a running Caudipteryx, we used classical aerodynamic equations to estimate the forces the wings would have produced when extended laterally in a fixed position during running at constant speed on level ground in still air. Only the two-dimensional shape of each wing, as opposed to its cross-sectional area, affected these calculations. Parameters and calculated results for this part of the analysis, assuming for the sake of example that Caudipteryx was running at its estimated maximum speed of 8 m/s, are given in Table S1.

In this situation, each wing would generate lift and drag according to Bernoulli’s principle8,18,19,20,21,22,23,24,25,26,27 (Fig. 2a,b), which may be expressed as

$${p}+\frac{1}{2}\rho {{v}}^{2}={\rm{Constant}}$$ (1)

where p is pressure, ρ is the density of the air, and v is the velocity of the air relative to the wing, which in still air corresponds to the velocity of the running animal. The constant term in this equation represents the total strength possessed by the air making up a particular streamline (i.e. line of flow) across the wing. The lift and drag forces are calculated as

$$\{\begin{array}{c}{{F}}_{{L}}=\frac{1}{2}\rho {{S}}_{{p}}{{C}}_{{L}}{{v}}^{2}\\ {{F}}_{{D}}=\frac{1}{2}{\rho }{{S}}_{{p}}{{C}}_{{D}}{{v}}^{2}\end{array}$$ (2)

where C L and C D represent coefficients of lift and drag, respectively, S P is the area of the wing projected into a plane perpendicular to the airflow (approximately half the true wing area, at an optimal angle of attack) and v represents airflow velocity. C L and C D cannot be precisely calculated based on the shape and material properties of a wing-like structure, but must generally be determined empirically or estimated based on reasonable assumptions28,29,30,31,32. For the wing of an extinct theropod, only the latter approach is applicable.

In the case of lift, we take C L to be equal to 2.0, a conservative value for the maximal lift coefficient (i.e. the lift coefficient associated with an optimal angle of attack) of a slotted wing19. This is appropriate because the gaps between the ends of the individual remiges on the forelimb of Caudipteryx would act aerodynamically as slots.

Determining a realistic value for C D is more complicated. The total drag experienced by each wing of Caudipteryx is the sum of induced drag, which occurs as a result of the same deflection of the airflow that produces lift, and profile drag, which results from a combination of friction with air in the laminar boundary layer passing over the wing surface and pressure exerted by air colliding with the wing as the latter moves forward. The coefficients associated with profile and induced drag, respectively C D(profile) and C D(induced) , can be calculated as19,32,33,34

$$\{\begin{array}{c}{{C}}_{{D}({profile})}\approx \frac{{{S}}_{{w}}}{{{S}}_{{p}}}\frac{1.33}{{\mathrm{Re}}^{0.5}}\approx \frac{2.6}{{\mathrm{Re}}^{0.5}}\\ {{C}}_{{D}({induced})}\approx \frac{{k}{{C}}_{{L}}^{2}}{\pi {A}}\end{array}$$ (3)

where S w is the area of the wing, A is the aspect ratio of the wing (A = 3.2, with the wing fully extended) and k is the induced drag factor, which is dependent on wing shape. Because the outline of the wing of Caudipteryx approximates an ellipse, we assign k a value of equal to π/4, appropriate for an elliptical wing. Re is the Reynolds number for the wing of Caudipteryx, representing the dimensionless ratio of inertial to viscous forces acting on the wing. Re may be calculated according to the standard formula Re = ρvl/η where l represents the length of the wing and η represents air viscosity. Assuming ρ = 1.21 kg/m3 and η = 1.8 × 10−5 Ns/m2, Re for the wing of Caudipteryx is 1.29 × 10−5, a value within the range for extant birds in flight. It is notable that C D(profile) will decrease as velocity increases, and therefore Reynolds number increases, too. The total drag coefficient for one wing of Caudipteryx is19,32,33,34,35,36,37

$${C}_{D}\approx {C}_{D({profile})}+{C}_{D({induced})}\approx \frac{2.6}{{\mathrm{Re}}^{0.5}}+\frac{k{C}_{{L}}^{2}}{\pi A}$$ (4)

Total drag force on the wing, F D , represents the sum of profile drag force F D(profile) and induced drag force F D(induced) , and may be written as

$$\{\begin{array}{c}{{F}}_{{D}({profile})}=\frac{1}{2}\rho {{S}}_{{p}}{{C}}_{{D}({profile})}{{v}}^{2}\Rightarrow {{F}}_{{D}({profile})}=\frac{1.33\rho {{S}}_{{p}}{{v}}^{2}}{{\mathrm{Re}}^{0.5}}\\ {{F}}_{{D}({induced})}=\frac{1}{2}\rho {{S}}_{{p}}{{C}}_{{D}({induced})}{{v}}^{2}\Rightarrow {{F}}_{{D}({induced})}=\frac{\rho {k}{{C}}_{{L}}^{2}{{S}}_{{p}}{{v}}^{2}}{2\pi {A}}\end{array}$$ (5)

$${{F}}_{{D}}={{F}}_{{D}({profile})}+{{F}}_{{D}({induced})}=\frac{1}{2}\rho {{S}}_{{p}}(\frac{2.6}{{\mathrm{Re}}^{0.5}}+\frac{{k}{{C}}_{{L}}^{2}}{\pi {A}}){{v}}^{2}$$ (6)

The total drag force on the wings and body of a moving Caudipteryx can then be expressed as19

$${{F}}_{D{,}\text{total}}=2[{{F}}_{{D}({profile})}+{{F}}_{{D}({induced})}]+{{F}}_{D{,}{body}}=\frac{1}{2}{\rho }{{v}}^{2}[2{{S}}_{{p}}(\frac{2.6}{{\mathrm{Re}}^{0.5}}+\frac{{k}{{C}}_{{L}}^{2}}{\pi {A}})+{{S}}_{{b}}{{C}}_{{D},{body}}]$$ (7)

where S b is the area of the front of the body projected into a transverse plane and C D(body) is the body drag coefficient. The term S b C D(body) can be replaced by the area of a flat plate transverse to the air stream that would produce an equivalent amount of drag, or A e . Tucker19 deduced from experiments on living birds the formula A e = (3.34 × 10−3)m2/3, where m is body mass (5 kg for Caudipteryx). Values of lift and total (i.e. wings and body) drag for different aspect ratio values from 0.4 to 3.2, reflecting different amounts of wing unfolding, are shown in Fig. 2c,d. Lift and drag curves in this figure were calculated for a range of running velocities, always assuming a headwind with an airspeed of 0.5 m/s and a lift coefficient of 2.038,39,40,41,42.

Theoretical analysis of a rectangular wing

In the next stage of our theoretical analysis, we modelled the interaction between the wing of Caudipteryx and the surrounding air in more concrete and physically explicit terms. We initially considered the wing of Caudipteryx to be a thin rectangular plate, which could be held extended laterally from the shoulder at varying angles of attack. The dimensions of the rectangle corresponded to the length and average chord of the real wing (0.24 m and 0.10 m respectively). We assumed the animal was running horizontally in a straight line at constant speed, with no air movement relative to the wing other than that produced by the forward motion of the body. We also ignored the effect of gravity on the wing. In the wing’s frame of reference, horizontally moving air would contact the lower surface of the wing and subsequently flow parallel to that surface, creating both vertical lift and horizontal drag (Fig. 3a,b).

The resultant of lift and drag, F w , would be a force vector directed backward and upward, acting perpendicular to the surface of the wing. F w is given by

$${{\boldsymbol{F}}}_{{w}}=\mathop{{m}}\limits^{\bullet }({{v}}_{2}-{{v}}_{1})$$ (8)

where m is the mass of air flow across the underside of the wing (i.e. the total mass of air that flows across this surface per second) and \(\mathop{{m}}\limits^{\bullet }=\frac{{dm}}{{dt}}\) is the derivative of mass with respect to time, and v 1 and v 2 respectively represent the velocity of incoming air as it contacts the leading edge of the wing and the velocity of outgoing air as it moves away from the trailing edge. The vector quantities v 1 and v 2 are equal in magnitude, but differ in direction because the airstream is reorientated as a result of contact with the wing. The mass flow is given by m = ρBv, indicating that mass flow is a function of the density of air (ρ air , in kg/m3), the area of the wing projected into a vertical plane (B, in m2), and velocity (v, in m/s, representing the common magnitude of v 1 and v 2 ). The vertical and horizontal components of F w , respectively approximating lift and drag, may be calculated according to:

$$\{\begin{array}{ccc}\sum {{F}}_{x} & = & 0\to {{F}}_{w(x)}+\mathop{{m}}\limits^{\bullet }({{v}}_{2(x)}-{{v}}_{1(x)})\\ & = & 0\to {{F}}_{w(x)}\\ & = & \mathop{{m}}\limits^{\bullet }({{v}}_{1(x)}-{{v}}_{2(x)})\to {{F}}_{w(x)}\\ & = & \mathop{{mv}}\limits^{\bullet }(1-{\rm{c}}{\rm{o}}{\rm{s}}\alpha )\\ \sum {{F}}_{y} & = & 0\Rightarrow {{F}}_{w(y)}+\mathop{{m}}\limits^{\bullet }({{v}}_{2(y)}-{{v}}_{1(y)})\\ & = & 0\to {{F}}_{w(y)}\\ & = & \mathop{{m}}\limits^{\bullet }({{v}}_{1(y)}-{{v}}_{2(y)})\to {{F}}_{w(y)}\\ & = & \mathop{{mv}}\limits^{\bullet }[0-(-{\rm{s}}{\rm{i}}{\rm{n}}\alpha )]\end{array}$$ (9)

where α is the angle of attack. Based on the expression given above for mass flow, the approximate vertical and horizontal aerodynamic forces on a hypothetical rectangular Caudipteryx wing are:

$$\{\begin{array}{c}{{F}}_{{w}({x})}=\rho {b}{{v}}^{2}(1-{\rm{c}}{\rm{o}}{\rm{s}}\alpha )\\ {{F}}_{{w}({y})}=\rho {b}{{v}}^{2}{\rm{s}}{\rm{i}}{\rm{n}}\alpha \end{array}$$ (10)

where b = wl sinα (m2), l = 0.100 m and w = 0.240 m.

In addition to this mathematical analysis, the fluid dynamics software package ABAQUS was used to simulate airflow patterns about the hypothetical flat rectangular wing, assuming the wing was held at various angles of attack during steady running at 8 m/s (Fig. S2).

Theoretical analysis of running at an angle of inclination with wings extended in a moving airstream

In this stage of the analysis we again assumed the wing to be a rectangular plate (AB in Fig. 4a). We considered the wing to be moving at a velocity v 1 , reflecting the velocity of the body of Caudipteryx in running either across level ground or on a slope, and encountering air moving in a horizontal or inclined direction at a velocity v 2 . In this situation the relative velocity of the airflow with respect to the wing, v, is the resultant of v 1 and v 2 : v 2 = v 2 – v 1 (Fig. 4a). The wing is held at an angle φ relative to the horizontal, but this is not necessarily equivalent to the angle of attack α because the airstream may not be moving in a horizontal direction. Accordingly, the equations used in this stage of the analysis offer a general quantitative description of the wing of Caudipteryx interacting with an airstream, assuming only that the wing extends sideways from the body (rather than being elevated or depressed, though it may be rotated about its long axis) and that both Caudipteryx and the airstream are constrained to a two-dimensional plane. This approach could therefore be used to evaluate the ability of the wing to produce aerodynamic forces helpful in executing manoeuvres such as braking and turning, although we did not explicitly pursue the issue in this study.

The amplitude of the relative velocity v can be calculated based on the cosine law of triangles:

$${v}=\sqrt{{({{v}}_{{x}}+{{v}}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta )}^{2}+{({{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}})}^{2}}$$ (11)

where v x and v y represent the horizontal and vertical components of the absolute speed of the Caudipteryx wing (i.e. the horizontal and vertical components of v 1 ), and θ is the angle of the airflow relative to the horizontal. According to the sine law of triangles:

$$\psi ={\rm{a}}{\rm{r}}{\rm{c}}{\rm{t}}{\rm{a}}{\rm{n}}(\frac{{{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}}}{{{v}}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta +{{v}}_{{x}}})$$ (12)

where ψ is the angle between v, the vector of airflow relative to the wing, and the horizontal. The angle of attack α, representing the angle between an equivalent chord AB (i.e. a line extending between the leading and trailing edges of the wing, in the plane of the wing surface) and the relative airflow velocity v is

$$\alpha =\phi +\psi =\phi +\arctan (\frac{{{v}}_{2}\,\sin \,\theta -{{v}}_{y}}{{{v}}_{2}\,\cos \,\theta +{{v}}_{x}})$$ (13)

where φ is the angle between the horizontal and the equivalent chord AB. Based on aerodynamic theory, the lift and drag exerted on the wing are then

$$\{\begin{array}{c}{{F}}_{{L}}=\frac{1}{2}{{C}}_{{L}}\rho {s}{{v}}^{2}=\frac{1}{2}{{C}}_{{L}}\rho {s}[{({{v}}_{{x}}+{{v}}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta )}^{2}+{({{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}})}^{2}]\\ {{F}}_{{D}}=\frac{1}{2}{{C}}_{{D}}\rho {s}{{v}}^{2}=\frac{1}{2}{{C}}_{{D}}\rho {s}[{({{v}}_{{x}}+{{v}}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta )}^{2}+{({{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}})}^{2}]\end{array}$$ (14)

where F L represents the magnitude of lift, acting perpendicular to the relative velocity v, and F D represents the magnitude of drag, acting opposite to the relative velocity v (Fig. 4a). C L and C D are the lift and drag coefficients of the Caudipteryx wing, while ρ represents the density of the moving air and s is the effective area of the wing. Therefore, the resultant force R acting on the wing is given by

$$\begin{array}{c}{\boldsymbol{R}}=[\begin{array}{c}{{R}}_{{x}}({{v}}_{{x}}{,}{{v}}_{{y}}{,}{{v}}_{2})\\ {{R}}_{{y}}({{v}}_{{x}}{,}{{v}}_{{y}}{,}{{v}}_{2})\end{array}]=[\begin{array}{c}{{F}}_{{D}}\,{\rm{c}}{\rm{o}}{\rm{s}}\,\psi -{{F}}_{{L}}\,{\rm{s}}{\rm{i}}{\rm{n}}\,\psi \\ {{F}}_{{D}}\,{\rm{s}}{\rm{i}}{\rm{n}}\,\psi +{{F}}_{{L}}\,{\rm{c}}{\rm{o}}{\rm{s}}\,\psi \end{array}]\\ {\boldsymbol{R}}=[\begin{array}{c}\frac{1}{2}\rho {s}[{({{v}}_{{x}}+{{v}}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta )}^{2}+{({{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}})}^{2}]({{C}}_{{D}}\,{\rm{c}}{\rm{o}}{\rm{s}}\,\psi -{{C}}_{{L}}\,{\rm{s}}{\rm{i}}{\rm{n}}\,\psi )\\ \frac{1}{2}\rho {s}[{({{v}}_{{x}}+{v}_{2}{\rm{c}}{\rm{o}}{\rm{s}}\theta )}^{2}+{({{v}}_{2}{\rm{s}}{\rm{i}}{\rm{n}}\theta -{{v}}_{{y}})}^{2}]({{C}}_{{D}}\,{\rm{s}}{\rm{i}}{\rm{n}}\,\psi +{{C}}_{{L}}\,{\rm{c}}{\rm{o}}{\rm{s}}\,\psi )\end{array}]\end{array}$$ (15)

This equation indicates that resultant force is dependent on the magnitude of the relative airflow v, as well as on the angle of inclination of the airflow θ and the lift and drag coefficients (C L and C D ).

The total power (P horizontal ) that the muscles of Caudipteryx must constantly supply in order to continue running at a given speed v is the dot product of v and the total drag on the animal, F D,total :

$${{P}}_{{horizontal}}={{\boldsymbol{F}}}_{{D},{total}}\cdot {\boldsymbol{v}}=\frac{1}{2}\rho {{v}}^{3}[2{{S}}_{{p}}(\frac{2.6}{{\mathrm{Re}}^{0.5}}+\frac{{k}{{C}}_{L}^{2}}{\pi {A}})+{{A}}_{{e}}]$$ (16)

As was the case in the earlier stage of our analysis, F D,total represents the sum of the drag on both wings and on the body.

The input metabolic power needed for the muscles to generate P horizontal would be P input = P horizontal /η, where η represents the efficiency of the muscles of Caudipteryx. Assuming 10% of total muscular power would have been devoted to functions other than locomotion at any given time, a more realistic computation of input power would be

$${{P}}_{{input}}=[1.1{P}]/\eta $$ (17)

The total metabolic power (i.e. input metabolic power) expended by Caudipteryx during terrestrial running is shown, for a range of velocities, in Fig. S5. In performing these metabolic power calculations, we assumed values of zero both for φ and for γ, the angle between the horizontal and the velocity vector of Caudipteryx v 1 . This implies that Caudipteryx was running on a horizontal substrate, with the wing also held horizontally. The angle of attack would then depend on the angle of the incident airflow v 2 . In this situation the aspect ratio of the wing, which would effectively vary with the degree of wing folding, would have an effect on the horizontal and vertical components of the resultant force (R x and R y ), which approximately correspond to drag and lift for small values of θ.

Analysis of forces generated by the wing in a series of positions corresponding to a downstroke

In order to produce aerodynamic thrust that might contribute to propulsion, a running Caudipteryx would need to flap its wings, not merely hold them extended in a fixed position relative to the body. Our analysis did not explicitly model flapping, but as a preliminary assessment of the aerodynamic forces associated with flapping wing movements we examined the effects of running with the wing held fixed in a series of positions approximating different stages in a downstroke (Fig. 5). A realistic 3D wing geometry was assumed in this part of the analysis, and Caudipteryx was considered to be running horizontally in still air at a speed of 8 m/s. Each downstroke position was defined by an angle of elevation or depression relative to the horizontal, which we term here the flapping angle β, and by twisting of the wing surface along its length, causing variation in the angle of attack across different parts of the wing. Twisting was modelled explicitly by dividing the wing along its length into a series of transversely narrow strips, with the angle of attack α changing incrementally from one strip to the next. The amount and direction of twisting were chosen, for each value of β, to produce a geometry corresponding to the shape the wing would be expected to adopt at that stage in a real downstroke. This part of analysis is thus physically unrealistic in that, during running with the wing held fixed at a particular β angle rather than actively moving up and down, no force would be available to produce the wing twisting that is being assumed to occur. However, we consider our attempt to model this situation to represent an informative thought as an experiment addressing the question of what forces the wing would be expected to produce if it could be held fixed with particular amounts of twisting at a range of β angle. Twisting of the wing makes possible the production of thrust, which acts in an opposite direction to drag. Accordingly, thrust and drag can be represented as a single thrust/drag vector acting on the wing, and pointing anteriorly if thrust exceeds drag but posteriorly if the opposite is the case.

We used the software package ABAQUS to estimate the aerodynamic forces associated with holding the wing fixed in each downstroke position. Because it is unlikely that Caudipteryx had a range of shoulder joint motion approaching that of modern birds, we considered only six modest flapping angles (β = −10°, −5°, 0°, 5°, 10° and 20°, with negative values corresponding to elevation and positive ones to depression) in this part of our analysis.

Figure 5a shows forces of lift and thrust/drag acting on the wing, in addition to their resultant force F w , for various points between the shoulder and wingtip. When averaged over the entire wing (Fig. 5b), F w normally includes three components: lift, thrust/drag, and a force acting along the mediolateral length of the wing. The magnitude and direction of the averaged F w , considered to act at the wing’s center of gravity, depend on the speed of the airstream in addition to wing 3D shape and wing position as expressed by flapping angle, angle of attack (at the base of the wing) and the lengthwise twisting of the wing’s surface. Drag tends to predominate over thrust near the base of the wing, but the reverse occurs near the tip, because the primary feathers play the main role in producing thrust (Fig. 5a).

Parameters simulated in this part of the analysis include velocity vectors for the airflow surrounding the wing (Fig. 6a,b); aerodynamic pressure over the top and bottom surfaces of the wing (Fig. 6c; Supplementary Videos 1 and 2); stresses on, as well as deflections of, the wing resulting from interaction with the airflow (Fig. 6d,e; Supplementary Videos 3 and 4); and lift and thrust/drag produced by the wing, shown as functions of the flapping angle (Fig. 5d). In Fig. 5d, interpolation based on results from the six discrete flapping angles tested in the analysis was used to produce continuous curves.

Experiments with a Caudipteryx robot

We constructed a robot (see Supplementary 3D Model) based on the skeletal and plumage anatomy of fossil specimens of Caudipteryx. The skeletal proportions of the robot were based mainly on BPM 0001, whereas the proportions and arrangement of the remiges were based mainly on IVPP V12430 and IVPP V12344. Most of the robot was built from ABS plastic, but the wing plumage was composed of trimmed feathers taken from modern birds (Fig. 7a). Because there is no evidence for tertiary feathers in Caudipteryx, feathers were attached only to the hand and forearm segments of the wing skeleton, using metal pins. An airflow from the anterior direction, relative to the body orientation of the Caudipteryx robot, was provided at speeds of 3.5 m/s and 6.0 m/s (see Supplementary Video 5). Aerodynamic forces exerted on the wings, namely lift and thrust/drag, were measured using sensors positioned at the wing bases. The shoulder joint was designed to minimize inertial and frictional effects during the experiment.